SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS


 Kerrie Randall
 5 years ago
 Views:
Transcription
1 SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering and D225 Dynamics of Mechanical Systems. On completion of this short tutorial you should be able to do the following. Describe a mechanism. Define relative and absolute velocity. Define relative and absolute acceleration. Define radial and tangential velocity. Define radial and tangential acceleration. Describe a four bar chain. Solve the velocity and acceleration of points within a mechanism. Use mathematical and graphical methods. Construct velocity and acceleration diagrams. Define the Coriolis Acceleration. Solve problems involving sliding links. It is assumed that the student is already familiar with the following concepts. Vector diagrams. Simple harmonic motion. Angular and linear motion. Inertia force. Appropriate level of mathematics. All these above may be found in the prerequisite tutorials. D.J.DUNN 1
2 1. INTRODUCTION A mechanism is used to produce mechanical transformations in a machine. This transformation could be any of the following. It may convert one speed to another speed. It may convert one force to another force. It may convert one torque to another torque. It may convert force into torque. It may convert one angular motion to another angular motion. It may convert angular motion into linear motion. It may convert linear motion into angular motion. A good example is a crank, connecting rod and piston mechanism. Figure 1 If the crank is turned, angular motion is converted into linear motion of the piston and input torque is transformed into force on the piston. If the piston is forced to move, the linear motion is converted into rotary motion and the force into torque. The piston is a sliding joint and this is called PRISMATIC in some fields of engineering such as robotics. The pin joints allow rotation of one part relative to another. These are also called REVOLUTE joints in other areas of engineering. Consider the next mechanism used in shaping machines and also known as the Whitworth quickreturn mechanism. Figure 2 D.J.DUNN 2
3 The input is connected to a motor turning at constant speed. This makes the rocking arm move back and forth and the head (that carries the cutting tool) reciprocates back and forth. Depending on the lengths of the various parts, the motion of the head can be made to move forwards at a fairly constant cutting speed but the return stroke is quick. Note that the pin and slider must be able to slide in the slot or the mechanism would jam. This causes problems in the solution because of the sliding link and this is covered later under Coriolis acceleration. The main point is that the motion produced is anything but simple harmonic motion and at any time the various parts of the mechanism have a displacement, velocity and acceleration. The acceleration gives rise to inertia forces and this puts stress on the parts in addition to the stress produced by the transmission of power. For example the acceleration of a piston in an internal combustion engine can be enormous and the connecting rod is subjected to high stresses as a result of the inertia as well as due to the power transmission. You will find in these studies that the various parts are referred to as links and it can be shown that all mechanisms are made up of a series of four links. The basic four bar link is shown below. When the input link rotates the output link may for example swing back and forth. Note that the fourth link is the frame of the machine and it is rigid and unable to move. With experience you should be able to identify the four bar chains in a mechanism. All the links shown are rigid links which means they may push or pull. It is possible to have links made of chain or rope which can only pull. Figure 3 D.J.DUNN 3
4 2. DISPLACEMENT, VELOCITY AND ACCELERATION All parts of a mechanism have displacement, velocity and acceleration. In the tutorial on free vibration, a mechanism called the Scotch Yoke was examined in order to explain sinusoidal or harmonic motion. The wheel turns at a constant speed and the yoke moves up and down. Figure 4 It was shown that the displacement x, velocity v and acceleration a of point p was given as follows. Angle θ = ωt Displacement x = R sin(ωt). Velocity v = dx/dt = ωr cos(ωt) Acceleration a = dv/dt = ω 2 R sin(ωt) The values can be calculated for any angle or moment of time. The acceleration could then be used to calculate the inertia force needed to accelerate and decelerate the link. Clearly it is the maximum values that are needed. Other mechanisms can be analysed mathematically in the same way but it is more difficult. The starting point is to derive the equation for displacement with respect to angle or time and then differentiate twice to get the acceleration. Without the aid of a computer to do this, the mathematics is normally much too difficult and a graphical method should be used as shown later. WORKED EXAMPLE No.1 A crank, con rod and piston mechanism is shown below. Determine the maximum acceleration of the piston when the crank speed is 30 rev/min clockwise. Figure 5 D.J.DUNN 4
5 SOLUTION When θ = 0 the piston will be furthest left at a distance of 170 mm from point O. Take this as the reference point and measure displacement x from there. Remember that θ = ωt and ω = 2π x 30/60 = rad/s. The displacement is then Differentiate to get the velocity Differentiate again to get the acceleration. The diagram shows a plot of displacement, velocity and acceleration against angle. It should be noted that none of them are sinusoidal and not harmonic (in particular, the acceleration). Figure 6 The units are all in mm and seconds. The above was done with a computer package. Plotting the above functions over a complete rotation shows that the maximum acceleration occurs at t = 0 (θ = 0) and evaluating gives an answer of 700 mm/s 2. D.J.DUNN 5
6 If the radius of the crank is small in comparison to the length of the connecting rod, the motion becomes close to sinusoidal. To illustrate this, here is the plot with the crank radius reduced to 10 mm. The acceleration is now almost a cosine curve. Figure 7 Solving these problems mathematically is difficult so we will now look at a graphical method. 3. VELOCITY DIAGRAMS This section involves the construction of diagrams which needs to be done accurately and to a suitable scale. Students should use a drawing board, ruler, compass, protractor and triangles and possess the necessary drawing skills. ABSOLUTE AND RELATIVE VELOCITY An absolute velocity is the velocity of a point measured from a fixed point (normally the ground or anything rigidly attached to the ground and not moving). Relative velocity is the velocity of a point measured relative to another that may itself be moving. TANGENTIAL VELOCITY Consider a link A B pinned at A and revolving about A at angular velocity ω. Point B moves in a circle relative to point A but its velocity is always tangential and hence at 90 o to the link. A convenient method of denoting this tangential velocity is (v B ) A meaning the velocity of B relative to A. This method is not always suitable. Figure 8 D.J.DUNN 6
7 RADIAL VELOCITY Consider a sliding link C that can slide on link AB. The direction can only be radial relative to point A as shown. If the link AB rotates about A at the same time then link C will have radial and tangential velocities. Figure 9 Note that both the tangential and radial velocities are denoted the same so the tags radial and tangential are added. The sliding link has two relative velocities, the radial and the tangential. They are normal to each other and the true velocity relative to A is the vector sum of both added as shown. Note that lower case letters are used on the vector diagrams. The two vectors are denoted by c 1 and c 2. The velocity of link C relative to point A is the vector a c 2. CRANK, CONNECTING ROD AND PISTON Figure 10 Consider this mechanism again. Let s freeze the motion (snap shot) at the position shown. The diagram is called a space diagram. Figure 11 D.J.DUNN 7
8 Every point on every link has a velocity through space. First we label the centre of rotation, often this is the letter O. Point A can only move in a tangential direction so the velocity of A relative to O is also its absolute velocity and the vector is normal to the crank and it is designated (v A ) O. (Note the rotation is anticlockwise). Now suppose that you are sat at point A and everything else moves relative to you. Looking towards B, it would appear the B is rotating relative to you (in reality it is you that is rotating) so it has a tangential velocity denoted (V B ) A. The direction is not always obvious except that it is normal to the link. Consider the fixed link OC. Since both points are fixed there is no velocity between them so so (v C ) O = 0 Next consider that you at point C looking at point B. Point B is a sliding link and will move in a straight line in the direction fixed by the slider guides and this is velocity (v B ) C. It follows that the velocity of B seen from O is the same as that seen from C so (v B ) C = (v B ) O The absolute velocity of B is (v B ) C = (v B ) O and this must be the vector sum of (V A ) O and (v B ) A and the three vectors must form a closed triangle as shown. The velocity of the piston must be in the direction in which it slides (conveniently horizontal here). This is a velocity diagram. METHODOLOGY Figure 12 First calculate the tangential velocity (v A ) O from v = ω x radius = ω x OA Draw the vector o  a in the correct direction (note lower case letters). We know that the velocity of B relative to A is to be added so the next vector ab starts at point a. At point a draw a line in the direction normal to the connecting rod but of unknown length. We know that the velocity of B relative and absolute to O is horizontal so the vector ob must start at a. Draw a horizontal line (in this case) through o to intersect with the other line. This is point b. The vectors ab and ob may be measured or calculated. Usually it is the velocity of the slider that is required. In a design problem, this velocity would be evaluated for many different positions of the crank shaft and the velocity of the piston determined for each position. Remember that the slider direction is not always horizontal and the direction of o  b must be the direction of sliding. D.J.DUNN 8
9 WORKED EXAMPLE No.2 The mechanism shown has a crank 50 mm radius which rotates at 2000 rev/min. Determine the velocity of the piston for the position shown. Also determine the angular velocity of link AB about A. SOLUTION Figure 13 Note the diagrams are not drawn to scale. The student should do this using a suitable scale for example 1 cm = 1 m/s. This is important so that the direction at 90 o to the link AB can be transferred to the velocity diagram. Angular speed of the crank ω = 2πN/60 = 2π x 2000/60 = rad/s (v A ) O = ω x radius = x 0.05 = m/s. First draw vector oa. (diagram a) Next add a line in the direction ab (diagram b) Finally add the line in the direction of ob to find point b and measure ob to get the velocity. (diagram C). Figure 14a Figure 14b Figure 14c The velocity of B relative to O is 7 m/s. The tangential velocity of B relative to A is the vector ab and this gives 9.2 m/s. The angular velocity of B about A is found by dividing by the radius (length of AB). ω for AB is then 9.2/0.09 = rad/s. (note this is relative to A and not an absolute angular velocity) D.J.DUNN 9
10 SELF ASSESSMENT EXERCISE No.1 Find the velocity of the piston for each case below and the angular velocity of AB about point A. 1. The crank OA rotates anticlockwise at 3000 rev/min. Answer 34 m/s and rad/s Figure The crank revolves clockwise at 300 rev/min. Note that the vector ob is still horizontal because the piston can only slide horizontally relative to O. Also the rotation of the crank is opposite to the previous cases so the direction of oa is down to the right. Figure 16 Answer 1.11 m/s to the right and 5.55 rad/s 3. The crank OA rotates at 200 rev/min clockwise. Note the vector ob is at 45 o to the horizontal as the piston must slide in this direction. Answer 0.49 m/s and 6.92 rad/s. Figure 17 D.J.DUNN 10
11 4 BAR CHAIN The input link rotates at a constant angular velocity ω 1. The relative velocity of each point relative to the other end of the link is shown. Each velocity vector is at right angles to the link. The output angular velocity is ω 2 and this will not be constant. The points A and D are fixed so they will appear as the same point on the velocity diagram. The methodology is the same as before and best shown with another example. Figure 18 WORKED EXAMPLE No. 3 Find the angular velocity of the output link when the input rotates at a constant speed of 500 rev/min. The diagram is not to scale. Figure 19 D.J.DUNN 11
12 SOLUTION First calculate ω 1. ω 1 = 2π x 500/60 = rad/s. Next calculate the velocity of point B relative to A. (V B ) A = ω 1 x AB = x 1 = m/s. Draw this as a vector to an appropriate scale. Figure 20a Next draw the direction of velocity C relative to B at right angles to the link BC passing through point b on the velocity diagram. Next draw the direction of the velocity of C relative to D at right angles to link DC passing through point a (which is the same as point d). Point c is where the two lines intersect, Figure 20b Figure 20c Determine velocity cd by measurement or any other method. The velocity of point C relative to D and is 43.5 m/s. Convert this into angular velocity by dividing the length of the link DC into it. ω 2 = 43.5/0.7 = 62 rad/s. D.J.DUNN 12
13 SELF ASSESSMENT EXERCISE No. 2 Determine the angular velocity of the link DC for each case shown and the direction of rotation. The diagrams are not to scale and should be constructed first. You are advised to use the best drawing instruments possible for accuracy. 1. The input rotates at 500 rev/min. Link BC is horizontal. (Ans. 76 rad/s clockwise.) Figure The input link AB rotates at 60 rev/min in a clockwise direction. (Ans. 16 rad/s) Figure 22 D.J.DUNN 13
14 4. ACCELERATION DIAGRAMS It is important to determine the acceleration of links because acceleration produces inertia forces in the link which stress the component parts of the mechanism. Accelerations may be relative or absolute in the same way as described for velocity. We shall consider two forms of acceleration, tangential and radial. Centripetal acceleration is an example of radial. CENTRIPETAL ACCELERATION A point rotating about a centre at radius R has a tangential velocity v and angular velocity ω and it is continually accelerating towards the centre even though it never moves any closer. This is centripetal acceleration and it is caused by the constant change in direction. It follows that the end of any rotating link will have a centripetal acceleration towards the opposite end. The relevant equations are: v = ωr a = ω 2 R or a = v 2 /R. The construction of the vector for radial acceleration causes confusion so the rules must be strictly followed. Consider the link AB. The velocity of B relative to A is tangential (v B ) A. The centripetal acceleration of B relative to A is in a radial direction so a suitable notation might be a R. It is calculated using a R = ω x AB or a R = v 2 /AB. Note the direction is towards the centre of rotation but the vector starts at a and ends at b 1. It is very important to get this the right way round otherwise the complete diagram will be wrong. Figure 23 D.J.DUNN 14
15 TANGENTIAL ACCELERATION Tangential acceleration only occurs if the link has an angular acceleration α rad/s 2. Consider a link AB with an angular acceleration about A. Figure 24 Point B will have both radial and tangential acceleration relative to point A. The true acceleration of point B relative to A is the vector sum of them. This will require an extra point. We will use b 1 and b on the vector diagram as shown. Point B is accelerating around a circular path and its direction is tangential (at right angles to the link). It is designated a T and calculated using a T = α x AB. The vector starts at b 1 and ends at b. The choice of letters and notation are arbitrary but must be logical to aid and relate to the construction of the diagram. Figure 25 D.J.DUNN 15
16 WORKED EXAMPLE No.4 A piston, connecting rod and crank mechanism is shown in the diagram. The crank rotates at a constant velocity of 300 rad/s. Find the acceleration of the piston and the angular acceleration of the link BC. The diagram is not drawn to scale. SOLUTION Figure 26 First calculate the tangential velocity of B relative to A. (v B ) A = ω x radius = 300 x 0.05 = 15 m/s. Next draw the velocity diagram and determine the velocity of C relative to B. From the velocity diagram (v C ) B = 7.8 m/s Figure 27 Next calculate all accelerations possible and construct the acceleration diagram to find the acceleration of the piston. The tangential acceleration of B relative to A is zero in this case since the link has no angular acceleration (α = 0). The centripetal acceleration of B relative to A a R = ω 2 x AB = x 0.05 = 4500 m/s 2. The tangential acceleration of C relative to B is unknown. The centripetal acceleration of C to B a R = v 2 /BC = /0.17 = m/s 2. The stage by stage construction of the acceleration diagram is as follows. D.J.DUNN 16
17 First draw the centripetal acceleration of link AB (Fig.a). There is no tangential acceleration so designate it ab. Note the direction is the same as the direction of the link towards the centre of rotation but is starts at a and ends at b. Figure 28a Figure 28b Figure 28c Next add the centripetal acceleration of link BC (Figure b). Since there are two accelerations for point C designate the point c 1. Note the direction is the same as the direction of the link towards the centre of rotation. Next add the tangential acceleration of point C relative to B (Figure c). Designate it c 1 c. Note the direction is at right angles to the previous vector and the length is unknown. Call the line a c line. Next draw the acceleration of the piston (figure d) which is constrained to be in the horizontal direction. This vector starts at a and must intersect the c line. Designate this point c. Figure 28d The acceleration of the piston is vector ac so (a C ) B = 1505 m/s 2. The tangential acceleration of C relative to B is c 1 c = 4000 m/s 2. At the position shown the connecting rod has an angular velocity and acceleration about its end even though the crank moves at constant speed. The angular acceleration of BC is the tangential acceleration divided by the length BC. α (BC) = 4000 / 0.17 = rad/s 2. D.J.DUNN 17
18 WORKED EXAMPLE No.5 The diagrams shows a rocking lever mechanism in which steady rotation of the wheel produces an oscillating motion of the lever OA. Both the wheel and the lever are mounted in fixed centres. The wheel rotates clockwise at a uniform angular velocity (ω) of 100 rad/s. For the configuration shown, determine the following. (i) The angular velocity of the link AB and the absolute velocity of point A. (ii) The centrifugal accelerations of BC, AB and OA. (iii) The magnitude and direction of the acceleration of point A. The lengths of the links are as follows. BC = 25 mm AB = 100 mm OA = 50 mm OC = 90 mm SOLUTION Figure 29 The solution is best done graphically. First draw a line diagram of the mechanism to scale. It should look like this. Figure 30 Next calculate the velocity of point B relative to C and construct the velocity diagram. D.J.DUNN 18
19 (v B ) C = ω x radius = 100 x = 2.5 m/s Figure 31 Scale the following velocities from the diagram. (v A ) O = 1.85 m/s {answer (i)} (v A ) B = 3.75 m/s Angular velocity = tangential velocity/radius For link AB, ω = 3.75/0.1 = 37.5 rad/s. {answer (i)} Next calculate all the accelerations possible. Radial acceleration of BC = ω 2 x BC = x = 250 m/s 2. {answer (ii)} Radial acceleration of AB = v 2 /AB = /0.1 = m/ s 2. {answer (ii)} Check same answer from ω 2 x AB = x 0.1 = m/ s 2. Radial Acceleration of OA is v 2 /OA = /0.05 = m/ s 2. {answer (ii)} Construction of the acceleration diagram gives the result shown. Figure 32 The acceleration of point A is the vector o a shown as a dotted line. Scaling this we get 560 m/s 2. {answer (iii)} D.J.DUNN 19
20 SELF ASSESSMENT EXERCISE No.3 Solve the acceleration of the piston for each case shown. You should draw the space diagram out accurately first. 1. (Ans. 153 m/s) Figure (Ans m/s 2 ) Figure 34 D.J.DUNN 20
21 WORKED EXAMPLE No. 6 Find the angular acceleration of the link CD for the case shown. SOLUTION Figure 35 First calculate or scale the length CB and find it to be 136 mm. Next find the velocities and construct the velocity diagram. Start with link AB as this has a known constant angular velocity. (v B ) A = ω x radius = 480 x 0.08 = 38.4 m/s Figure 36 Next calculate all the accelerations possible. The centripetal acceleration of B to A is /0.08 = m/s 2 The centripetal acceleration of C to D is 15 2 /0.16 = 1406 m/s 2 The centripetal acceleration of C to B is 31 2 /0.136 = 7066 m/s 2. We cannot calculate any tangential accelerations at this stage. The stage by stage construction of the acceleration diagram follows. D.J.DUNN 21
22 First draw the centripetal acceleration of B to A (Figure a). There is no tangential to add on). Figure 37a Figure 37b Figure 37c Next add the centripetal acceleration of C to B (figure b) Next draw the direction of the tangential acceleration of C to B of unknown length at right angles to the previous vector (figure c). Designate it as a c line. We cannot proceed from this point unless we realise that points a and d are the same (there is no velocity or acceleration of D relative to A). Add the centripetal acceleration of C to D (figure d). This is 1406 m/s 2 in the direction of link CD. Designte it d c 2. Figure 37d Figure 37e Finally draw the tangential acceleration of C to D at right angles to the previous vector to intersect the c line (figure e). From the diagram determine c 2 c to be m/s 2. This is the tangential acceleration of C to D. The angular acceleration of the link DC is then: α (CD) = 24000/0.16 = rad/s2 in a clockwise direction. Note that although the link AB rotates at constant speed, the link CD has angular acceleration. D.J.DUNN 22
23 WORKED EXAMPLE No. 7 The same arrangement exists as shown for example 5 except that the link AB is decelerating at 8000 rad/s 2 (i.e. in an anticlockwise direction). Determine the acceleration of the link CD. SOLUTION The problem is essentially the same as example 5 except that a tangential acceleration now exists for point B relative to point A. This is found from a T = α x AB = x 0.08 = 6400 m/s 2 T he direction is for an anticlockwise tangent. This is vector b1 b which is at right angles to a b 1 in the appropriate direction. The new acceleration diagram looks like this. Figure 38 Scaling off the tangential acceleration c 2 c we get m/s 2. Converting this into the angular acceleration we get α = /0.16 = rad/s 2 in a clockwise direction. D.J.DUNN 23
24 SELF ASSESSMENT EXERCISE No.4 1. The diagram shows a 4 bar chain. The link AB rotates at a constant speed of 5 rad/s in an anticlockwise direction. For the position shown, determine the angular acceleration of the link DC. Figure 39 (Answer 30 rad/s2 in an anticlockwise direction) 2. Repeat question 1 but this time the link AB is accelerating at 15 rad/s 2. (Answer 15.3 rad/s 2 in an anticlockwise direction) 3. The diagram shows the instantaneous position of a mechanism in which member OA rotates anticlockwise with an angular velocity of 100 rad/s and angular acceleration of rad/s 2 in the same direction. BD is a continuation of the rigid link AB. The links have the following lengths. OA 30 mm BC 90 mm AD 168 mm AB 1120 mm Determine the linear the following. i. The velocities of points A, B and D (1.5 m/s, 2.6 m/s and 2.7 m/s) ii. The absolute linear accelerations of points A and B ( m/s 2 and 440 m/s 2 ) Figure 40 D.J.DUNN 24
25 5. INERTIA FORCE One of the reasons for finding the acceleration of links is to calculate the inertia force needed to accelerate or decelerate it. This is based on Newton s second law. Force = mass x acceleration F = M a And Torque = moment of inertia x angular acceleration T = I α WORKED EXAMPLE No.8 A horizontal single cylinder reciprocating engine has a crank OC of radius 40 mm and a connecting rod PC 140 mm long as shown. The crank rotates at 3000 rev/min clockwise. For the configuration shown, determine the velocity and acceleration of the piston. The sliding piston has a mass of 0.5 kg and a diameter of 80 mm. The gas pressure acting on it is 1.2 MPa at the moment shown. Calculate the effective turning moment acting on the crank. Assume that the connecting rod and crank has negligible inertia and friction. SOLUTION Figure 41 Draw the space diagram to scale. Figure 42 The moment arm should be scaled and found to be 34 mm (measured at right angles to the connecting rod PC. Calculate the velocity of C relative to O. ω = 2πN/60 = 2π x 3000/60 = rad/s (v C ) O = ω x radius = x 0.04 = m/s D.J.DUNN 25
26 Draw the velocity diagram. Figure 43 From the velocity diagram we find the velocity of the piston is 11 m/s. Next calculate all the accelerations possible. Point C only has a radial acceleration towards O Radial acceleration of C is v 2 /radius = /0.04 = 3950 m/s 2 Point P has radial and tangential acceleration relative to C. Tangential acceleration is unknown. Radial acceleration = (v P ) C 2 /CP = 9 2 /0.14 = m/s 2 Now draw the acceleration diagram and it comes out like this. The acceleration of the piston is 2839 m/s 2. Now we can solve the forces. Figure 44 Pressure force = p x area = 1.2 x 10 6 x π x /4 = 6032 N and this acts left to right. Inertia force acting on the piston = M a = 0.5 x 2839 = N and this must be provided by the pressure force so the difference is the force exerted on the connecting rod. Net Force = = N. D.J.DUNN 26
27 The connecting rod makes an angle of 11 o to the line of the force (angle scaled from space diagram). This must be resolved to find the force acting along the line of the connecting rod. Figure 45 The force in the connecting rod is cos 11 o = 4528 N. This acts at a radius of 34 mm from the centre of the crank so the torque provided by the crank is T = 4528 x = 154 N m. SELF ASSESSMENT EXERCISE No.5 1. The piston in the mechanism shown has a mass of 0.8 kg. Determine its acceleration and the inertia force needed for the position shown. Figure 46 (Ans m/s 2 and 3200 N) D.J.DUNN 27
28 6. CORIOLIS ACCELERATION Consider a link rotating at ω rad/s and accelerating at α rad/s 2. On the link is a sliding element moving away from the centre of rotation at velocity v R = dr/dt (positive if getting larger) The link has a tangential velocity v T = ωr The component of this velocity in the x direction is v T sin θ = v T sin ωr The velocity v R also has a component in the x direction And this is v R cos θ The total velocity in the x direction of the sliding link is v x = v T sin θ + v R cos θ v x = (dr/dt) cos ωt + ωr sin ωt The acceleration in the x direction is a x Figure 47 2 dv d R dr 2 dt 2 dt dt a x = x =  cos( ω t) + ω sin(ω t) + ω R cos( ω t) + ω sin( ω t) + R sin( ω t) 2 d R dr 2 2 dt dt dω dt dr dt a x = cos( ω t) + 2 ω sin( ω t) + Rω cos( ω t) + R sin( ω t) 2 d 2 R dr a x = cos θ + 2 ω sin θ + R ω 2 dt dt When θ = 90 o dr a x = a T = 2 ω + α R dt a T = 2 ω vr + α R cosθ + α R sin θ The tangential acceleration is not simply α R as is the case for a constant radius but an extra term of 2ωvR is added and this term is called the Coriolis acceleration and must be taken into consideration when solving problems with changing radius. dω dt D.J.DUNN 28
29 WORKED EXAMPLE No.9 Figure 48 The diagram shows part of a quick return mechanism. The pin A slides in the slot when the disc is rotated. Calculate the angular velocity and acceleration of link BC when θ = 60 o and ω = 100 rad/s. SOLUTION The tangential velocity of A relative to O is ωr = 100 x 0.04 = 4 m/s. The velocity diagram is constructed as shown. Figure 49 The tangential velocity of pin A relative to B is (V A1 ) B = a a 1 = 4 cos (38.95 o ) = 3.11 m/s The radial velocity of A relative to B is (V A ) B = 4 sin(38.95 o ) = m/s The length of BA is easily calculated from the diagram. BA = ( ) = mm Figure 50 The angular velocity link BC = 3.11/BA = 32.2 rad/s D.J.DUNN 29
30 ANALYTICAL METHOD The angle of link BC is 1 40sin θ 1 sin θ α = tan = tan cos θ 7/4 + cos θ The angular velocity is dα/dt and the tools for doing the differentiation are given in the question as follows. sin θ dα 1 1 dx 1+ 7/4cos θ Let x = = = = 7/4 + cos θ 2 2 dx 1+ x sin θ dθ ( 7/4 + cos θ ) /4 + cos θ dα dα dx /4cos θ = = dt dx dθ sin θ 2 ( 7/4 cos θ ) 2 put θ = 60 o and evaluate /4 + cos θ dα/dθ = θ = ω t so dθ = ω dt answer found before. dt = dθ/ω so dα/dt = x ω = 31.6 rad/s which is close to the Next construct the acceleration diagram. Figure 51 Link O A only has centripetal acceleration inwards (a A ) O = ω 2 R = x 0.04 = 400 m/s 2 The pin A has a tangential acceleration and Coriolis acceleration normal to the link. It has centripetal acceleration and radial acceleration towards the centre of rotation. B. The diagram can be constructed without calculating them. The Corioilis acceleration is 2 ω v where ω = 32.2 and v is the radial velocity = m/s The Coriolis term is hence m/s 2 The tangential acceleration of A relative to B is a 1 a = 400 sin = m/s 2 Part of this is the Coriolis so the tangential acceleration is = m/s 2 The angular acceleration of link AC is α = /BA = 89.49/ = 928 rad/s 2 The direction is negative (clockwise) so it is decelerating. D.J.DUNN 30
31 SELF ASSESSMENT EXERCISE No.6 A link OA is 80 mm long and rotates at a constant speed of 50 rad/s. A sliding link attached to it slides on link BC and makes BC rotate about B as shown. Calculate the angular velocity and acceleration of BC when angle θ = 70 o. (22.8 rad/s and rad/s 2 ) Figure 52 D.J.DUNN 31
ANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME  TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More information11. Rotation Translational Motion: Rotational Motion:
11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational
More informationENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION
ENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION This tutorial covers prerequisite material and should be skipped if you are
More informationPHY121 #8 Midterm I 3.06.2013
PHY11 #8 Midterm I 3.06.013 AP Physics Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension
More informationSOLID MECHANICS DYNAMICS TUTORIAL MOMENT OF INERTIA. This work covers elements of the following syllabi.
SOLID MECHANICS DYNAMICS TUTOIAL MOMENT OF INETIA This work covers elements of the following syllabi. Parts of the Engineering Council Graduate Diploma Exam D5 Dynamics of Mechanical Systems Parts of the
More informationPHYSICS 111 HOMEWORK SOLUTION #9. April 5, 2013
PHYSICS 111 HOMEWORK SOLUTION #9 April 5, 2013 0.1 A potter s wheel moves uniformly from rest to an angular speed of 0.16 rev/s in 33 s. Find its angular acceleration in radians per second per second.
More informationChapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.
Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems
More informationLecture Presentation Chapter 7 Rotational Motion
Lecture Presentation Chapter 7 Rotational Motion Suggested Videos for Chapter 7 Prelecture Videos Describing Rotational Motion Moment of Inertia and Center of Gravity Newton s Second Law for Rotation Class
More informationSOLID MECHANICS BALANCING TUTORIAL BALANCING OF ROTATING BODIES
SOLID MECHANICS BALANCING TUTORIAL BALANCING OF ROTATING BODIES This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 4. On completion of this tutorial
More informationMechanical Principles
Unit 4: Mechanical Principles Unit code: F/601/1450 QCF level: 5 Credit value: 15 OUTCOME 4 POWER TRANSMISSION TUTORIAL 2 BALANCING 4. Dynamics of rotating systems Single and multilink mechanisms: slider
More informationwww.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x
Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity
More informationSlide 10.1. Basic system Models
Slide 10.1 Basic system Models Objectives: Devise Models from basic building blocks of mechanical, electrical, fluid and thermal systems Recognize analogies between mechanical, electrical, fluid and thermal
More informationChapter 3.8 & 6 Solutions
Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled
More informationCentripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.
Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.
More informationSolution Derivations for Capa #11
Solution Derivations for Capa #11 1) A horizontal circular platform (M = 128.1 kg, r = 3.11 m) rotates about a frictionless vertical axle. A student (m = 68.3 kg) walks slowly from the rim of the platform
More informationPhysics 201 Homework 8
Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 Nm is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kgm 2. What is the
More informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 13, 2014 1:00PM to 3:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of
More informationUnit 4 Practice Test: Rotational Motion
Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle
More information3600 s 1 h. 24 h 1 day. 1 day
Week 7 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationPhysics 2A, Sec B00: Mechanics  Winter 2011 Instructor: B. Grinstein Final Exam
Physics 2A, Sec B00: Mechanics  Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry
More informationPower Electronics. Prof. K. Gopakumar. Centre for Electronics Design and Technology. Indian Institute of Science, Bangalore.
Power Electronics Prof. K. Gopakumar Centre for Electronics Design and Technology Indian Institute of Science, Bangalore Lecture  1 Electric Drive Today, we will start with the topic on industrial drive
More informationAngular acceleration α
Angular Acceleration Angular acceleration α measures how rapidly the angular velocity is changing: Slide 70 Linear and Circular Motion Compared Slide 7 Linear and Circular Kinematics Compared Slide 7
More information3 Work, Power and Energy
3 Work, Power and Energy At the end of this section you should be able to: a. describe potential energy as energy due to position and derive potential energy as mgh b. describe kinetic energy as energy
More informationChapter 6 Circular Motion
Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example
More informationHW Set VI page 1 of 9 PHYSICS 1401 (1) homework solutions
HW Set VI page 1 of 9 1030 A 10 g bullet moving directly upward at 1000 m/s strikes and passes through the center of mass of a 5.0 kg block initially at rest (Fig. 1033 ). The bullet emerges from the
More informationChapter 11 Equilibrium
11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of
More informationMECHANICAL PRINCIPLES OUTCOME 4 MECHANICAL POWER TRANSMISSION TUTORIAL 1 SIMPLE MACHINES
MECHANICAL PRINCIPLES OUTCOME 4 MECHANICAL POWER TRANSMISSION TUTORIAL 1 SIMPLE MACHINES Simple machines: lifting devices e.g. lever systems, inclined plane, screw jack, pulley blocks, Weston differential
More informationLab 7: Rotational Motion
Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME9472), string with loop at one end and small white bead at the other end (125
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3D We have defined the velocit and acceleration of a particle as the first and second
More informationMechanical Principles
Unit 4: Mechanical Principles Unit code: F/60/450 QCF level: 5 Credit value: 5 OUTCOME 3 POWER TRANSMISSION TUTORIAL BELT DRIVES 3 Power Transmission Belt drives: flat and vsection belts; limiting coefficient
More informationSOLID MECHANICS DYNAMICS TUTORIAL PULLEY DRIVE SYSTEMS. This work covers elements of the syllabus for the Edexcel module HNC/D Mechanical Principles.
SOLID MECHANICS DYNAMICS TUTORIAL PULLEY DRIVE SYSTEMS This work covers elements of the syllabus for the Edexcel module HNC/D Mechanical Principles. On completion of this tutorial you should be able to
More informationPractice Exam Three Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004 Practice Exam Three Solutions Problem 1a) (5 points) Collisions and Center of Mass Reference Frame In the lab frame,
More informationKINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES
KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,
More informationTorque and Rotary Motion
Torque and Rotary Motion Name Partner Introduction Motion in a circle is a straightforward extension of linear motion. According to the textbook, all you have to do is replace displacement, velocity,
More informationPHYS 211 FINAL FALL 2004 Form A
1. Two boys with masses of 40 kg and 60 kg are holding onto either end of a 10 m long massless pole which is initially at rest and floating in still water. They pull themselves along the pole toward each
More informationPhysics 1A Lecture 10C
Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. Oprah Winfrey Static Equilibrium
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More informationSOLID MECHANICS DYNAMICS TUTORIAL CENTRIPETAL FORCE
SOLID MECHANICS DYNAMICS TUTORIAL CENTRIPETAL FORCE This work coers elements of the syllabus for the Engineering Council Exam D5 Dynamics of Mechanical Systems C10 Engineering Science. This tutorial examines
More informationGear Trains. Introduction:
Gear Trains Introduction: Sometimes, two or more gears are made to mesh with each other to transmit power from one shaft to another. Such a combination is called gear train or train of toothed wheels.
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1  LOADING SYSTEMS
EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1  LOADING SYSTEMS TUTORIAL 1 NONCONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects
More informationPHYS 1014M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
PHYS 1014M, Fall 2005 Exam #3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A bicycle wheel rotates uniformly through 2.0 revolutions in
More informationProblem Set 1. Ans: a = 1.74 m/s 2, t = 4.80 s
Problem Set 1 1.1 A bicyclist starts from rest and after traveling along a straight path a distance of 20 m reaches a speed of 30 km/h. Determine her constant acceleration. How long does it take her to
More informationVELOCITY, ACCELERATION, FORCE
VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how
More informationRotational Inertia Demonstrator
WWW.ARBORSCI.COM Rotational Inertia Demonstrator P33545 BACKGROUND: The Rotational Inertia Demonstrator provides an engaging way to investigate many of the principles of angular motion and is intended
More informationMidterm Exam 1 October 2, 2012
Midterm Exam 1 October 2, 2012 Name: Instructions 1. This examination is closed book and closed notes. All your belongings except a pen or pencil and a calculator should be put away and your bookbag should
More informationLinear Motion vs. Rotational Motion
Linear Motion vs. Rotational Motion Linear motion involves an object moving from one point to another in a straight line. Rotational motion involves an object rotating about an axis. Examples include a
More informationPHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013
PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be
More informationcircular motion & gravitation physics 111N
circular motion & gravitation physics 111N uniform circular motion an object moving around a circle at a constant rate must have an acceleration always perpendicular to the velocity (else the speed would
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationSOLID MECHANICS DYNAMICS TUTORIAL NATURAL VIBRATIONS ONE DEGREE OF FREEDOM
SOLID MECHANICS DYNAMICS TUTORIAL NATURAL VIBRATIONS ONE DEGREE OF FREEDOM This work covers elements of the syllabus for the Engineering Council Exam D5 Dynamics of Mechanical Systems, C05 Mechanical and
More informationCHAPTER 15 FORCE, MASS AND ACCELERATION
CHAPTER 5 FORCE, MASS AND ACCELERATION EXERCISE 83, Page 9. A car initially at rest accelerates uniformly to a speed of 55 km/h in 4 s. Determine the accelerating force required if the mass of the car
More informationLecture L5  Other Coordinate Systems
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5  Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates
More informationChapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.
Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular
More informationLecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is
Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.49.6, 10.110.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of
More informationLecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6
Lecture 16 Newton s Second Law for Rotation Moment of Inertia Angular momentum Cutnell+Johnson: 9.4, 9.6 Newton s Second Law for Rotation Newton s second law says how a net force causes an acceleration.
More informationMidterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m
Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is nonnegative
More informationPhysics Midterm Review Packet January 2010
Physics Midterm Review Packet January 2010 This Packet is a Study Guide, not a replacement for studying from your notes, tests, quizzes, and textbook. Midterm Date: Thursday, January 28 th 8:1510:15 Room:
More informationChapter 8: Rotational Motion of Solid Objects
Chapter 8: Rotational Motion of Solid Objects 1. An isolated object is initially spinning at a constant speed. Then, although no external forces act upon it, its rotational speed increases. This must be
More informationPhysics 1120: Simple Harmonic Motion Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured
More informationUnit 24: Applications of Pneumatics and Hydraulics
Unit 24: Applications of Pneumatics and Hydraulics Unit code: J/601/1496 QCF level: 4 Credit value: 15 OUTCOME 2 TUTORIAL 3 HYDRAULIC AND PNEUMATIC MOTORS The material needed for outcome 2 is very extensive
More informationCenter of Gravity. We touched on this briefly in chapter 7! x 2
Center of Gravity We touched on this briefly in chapter 7! x 1 x 2 cm m 1 m 2 This was for what is known as discrete objects. Discrete refers to the fact that the two objects separated and individual.
More informationTennessee State University
Tennessee State University Dept. of Physics & Mathematics PHYS 2010 CF SU 2009 Name 30% Time is 2 hours. Cheating will give you an Fgrade. Other instructions will be given in the Hall. MULTIPLE CHOICE.
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5  ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4  ALTERNATING CURRENT
EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5  ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4  ALTERNATING CURRENT 4 Understand singlephase alternating current (ac) theory Single phase AC
More informationLecture L222D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L  D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L3 for
More informationChosen problems and their final solutions of Chap. 2 (Waldron) Par 1
Chosen problems and their final solutions of Chap. 2 (Waldron) Par 1 1. In the mechanism shown below, link 2 is rotating CCW at the rate of 2 rad/s (constant). In the position shown, link 2 is horizontal
More informationPhysics 160 Biomechanics. Angular Kinematics
Physics 160 Biomechanics Angular Kinematics Questions to think about Why do batters slide their hands up the handle of the bat to lay down a bunt but not to drive the ball? Why might an athletic trainer
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More information226 Chapter 15: OSCILLATIONS
Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion
More informationOrbital Mechanics. Angular Momentum
Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely
More informationUniversal Law of Gravitation
Universal Law of Gravitation Law: Every body exerts a force of attraction on every other body. This force called, gravity, is relatively weak and decreases rapidly with the distance separating the bodies
More informationCoriolis acceleration
Alpha Omega Engineering, Inc. 872 Toro Street, San Luis Obispo, CA 93401 (805) 541 8608 (home office), (805) 441 3995 (cell) Coriolis acceleration by Frank Owen, PhD, P.E., www.aoengr.com All rights reserved
More informationFluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur
Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture  20 Conservation Equations in Fluid Flow Part VIII Good morning. I welcome you all
More informationElectric Motors and Drives
EML 2322L MAE Design and Manufacturing Laboratory Electric Motors and Drives To calculate the peak power and torque produced by an electric motor, you will need to know the following: Motor supply voltage,
More informationSample Questions for the AP Physics 1 Exam
Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiplechoice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationDesign of a Universal Robot Endeffector for Straightline Pickup Motion
Session Design of a Universal Robot Endeffector for Straightline Pickup Motion Gene Y. Liao Gregory J. Koshurba Wayne State University Abstract This paper describes a capstone design project in developing
More informationUnit  6 Vibrations of Two Degree of Freedom Systems
Unit  6 Vibrations of Two Degree of Freedom Systems Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Introduction A two
More informationLecture L6  Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6  Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationENGINEERING COUNCIL CERTIFICATE LEVEL ENGINEERING SCIENCE C103 TUTORIAL 3  TORSION
ENGINEEING COUNCI CETIFICATE EVE ENGINEEING SCIENCE C10 TUTOIA  TOSION You should judge your progress by completing the self assessment exercises. These may be sent for marking or you may request copies
More informationSpring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations
Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationChapter 18 Static Equilibrium
Chapter 8 Static Equilibrium 8. Introduction Static Equilibrium... 8. Lever Law... Example 8. Lever Law... 4 8.3 Generalized Lever Law... 5 8.4 Worked Examples... 7 Example 8. Suspended Rod... 7 Example
More information8.012 Physics I: Classical Mechanics Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE
More information11. Describing Angular or Circular Motion
11. Describing Angular or Circular Motion Introduction Examples of angular motion occur frequently. Examples include the rotation of a bicycle tire, a merrygoround, a toy top, a food processor, a laboratory
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationAP Physics C. Oscillations/SHM Review Packet
AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete
More informationPES 1110 Fall 2013, Spendier Lecture 27/Page 1
PES 1110 Fall 2013, Spendier Lecture 27/Page 1 Today:  The Cross Product (3.8 Vector product)  Relating Linear and Angular variables continued (10.5)  Angular velocity and acceleration vectors (not
More information14 Engineering physics
Option B 14 Engineering physics ESSENTIAL IDEAS The basic laws of mechanics have an extension when equivalent principles are applied to rotation. Actual objects have dimensions and they require an expansion
More informationChapter 11. h = 5m. = mgh + 1 2 mv 2 + 1 2 Iω 2. E f. = E i. v = 4 3 g(h h) = 4 3 9.8m / s2 (8m 5m) = 6.26m / s. ω = v r = 6.
Chapter 11 11.7 A solid cylinder of radius 10cm and mass 1kg starts from rest and rolls without slipping a distance of 6m down a house roof that is inclined at 30 degrees (a) What is the angular speed
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com
Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations
More informationLab 8: Ballistic Pendulum
Lab 8: Ballistic Pendulum Equipment: Ballistic pendulum apparatus, 2 meter ruler, 30 cm ruler, blank paper, carbon paper, masking tape, scale. Caution In this experiment a steel ball is projected horizontally
More informationProblem 6.40 and 6.41 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITSPilani
Problem 6.40 and 6.4 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITSPilani 6.40 A wheel with fine teeth is attached to the end of a spring with constant k and unstretched length
More informationThe Effects of Wheelbase and Track on Vehicle Dynamics. Automotive vehicles move by delivering rotational forces from the engine to
The Effects of Wheelbase and Track on Vehicle Dynamics Automotive vehicles move by delivering rotational forces from the engine to wheels. The wheels push in the opposite direction of the motion of the
More informationSimple Harmonic Motion
Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights
More informationBedford, Fowler: Statics. Chapter 4: System of Forces and Moments, Examples via TK Solver
System of Forces and Moments Introduction The moment vector of a force vector,, with respect to a point has a magnitude equal to the product of the force magnitude, F, and the perpendicular distance from
More informationLecture 07: Work and Kinetic Energy. Physics 2210 Fall Semester 2014
Lecture 07: Work and Kinetic Energy Physics 2210 Fall Semester 2014 Announcements Schedule next few weeks: 9/08 Unit 3 9/10 Unit 4 9/15 Unit 5 (guest lecturer) 9/17 Unit 6 (guest lecturer) 9/22 Unit 7,
More informationSolution: Angular velocity in consistent units (Table 8.1): 753.8. Velocity of a point on the disk: Rate at which bits pass by the read/write head:
Problem P8: The disk in a computer hard drive spins at 7200 rpm At the radius of 0 mm, a stream of data is magnetically written on the disk, and the spacing between data bits is 25 μm Determine the number
More information