Physics 101 Lecture 12 Equilibrium and Angular Momentum

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1 Physics 101 Lecture 1 Equilibrium and Angular Momentum Ali ÖVGÜN EMU Physics Department

2 Static Equilibrium q Equilibrium and static equilibrium q Static equilibrium conditions n Net external orce must equal zero n Net external torque must equal zero q Center o gravity q Solving static equilibrium problems /13/17

3 Static and Dynamic Equilibrium q Equilibrium implies the object is at rest (static) or its center o mass moves with a constant velocity (dynamic) q We will consider only with the case in which linear and angular velocities are equal to zero, called static equilibrium : v CM 0 and ω 0 q Examples n n n n n Book on table Hanging sign Ceiling an o Ceiling an on Ladder leaning against wall /13/17

4 Conditions or Equilibrium q The irst condition o equilibrium is a statement o translational equilibrium q The net external orce on the object must equal zero! F net!! F ma ext 0 q It states that the translational acceleration o the object s center o mass must be zero /13/17

5 Conditions or Equilibrium q I the object is modeled as a particle, then this is the only condition that must be satisied!! F net Fext 0 q For an extended object to be in equilibrium, a second condition must be satisied q This second condition involves the rotational motion o the extended object /13/17

6 Conditions or Equilibrium q The second condition o equilibrium is a statement o rotational equilibrium q The net external torque on the object must equal zero! τ net!! τ Iα 0 ext q It states the angular acceleration o the object to be zero q This must be true or any axis o rotation /13/17

7 Conditions or Equilibrium qthe net orce equals zero F r 0 n I the object is modeled as a particle, then this is the only condition that must be satisied qthe net torque equals zero r τ 0 n This is needed i the object cannot be modeled as a particle qthese conditions describe the rigid objects in the equilibrium analysis model /13/17

8 Static Equilibrium q (A) (B) (C) (D) Consider a light rod subject to the two orces o equal magnitude as shown in igure. Choose the correct statement with regard to this situation: The object is in orce equilibrium but not torque equilibrium. The object is in torque equilibrium but not orce equilibrium The object is in both orce equilibrium and torque equilibrium The object is in neither orce equilibrium nor torque equilibrium /13/17

9 Equilibrium Equations q For simplicity, We will restrict the applications to situations in which all the orces lie in the xy plane. q Equation 1: q Equation :! F! τ net net! Fext 0 : Fnet x 0 Fnet, y 0 Fnet, z! τ : τ 0 τ 0 τ, ext 0 net, x net, y net, z q There are three resulting equations F F 0 F τ net, x net, y net, z τ F ext, x ext, y ext, z /13/17

10 q EX: A seesaw consisting o a uniorm board o mass m pl and length L supports at rest a ather and daughter with masses M and m, respectively. The support is under the center o gravity o the board, the ather is a distance d rom the center, and the daughter is a distance.00 m rom the center. q A) Find the magnitude o the upward orce n exerted by the support on the board. q B) Find where the ather should sit to balance the system at rest. /13/17

11 A) Find the magnitude o the upward orce n exerted by the support on the board. B) Find where the ather should sit to balance the system at rest. F n τ x net, y net, z mgd n mg Mg m g 0 mg mgd τ m M + Mg + m d d + τ m M pl + τ g < + τ pl Mgx Mgx pl n.00 m F F τ net, x net, y net, z F F τ ext, x ext, y ext, z /13/17

12 Axis o Rotation q The net torque is about an axis through any point in the xy plane q Does it matter which axis you choose or calculating torques? q NO. The choice o an axis is arbitrary q I an object is in translational equilibrium and the net torque is zero about one axis, then the net torque must be zero about any other axis q We should be smart to choose a rotation axis to simpliy problems /13/17

13 B) Find where the ather should sit to balance the system at rest. Rotation axis O τ mgd x net, z m M d mgd Mgx τ + τ + τ + τ Mgx d m M pl n Rotation axis P τ 0 Mg( d Mgd Mgx m mgd x net, z τ + τ + τ + τ m M d Mgx d m M pl + x) m pl pl n gd + nd 0 gd + ( Mg + mg + m pl g) d 0 P O F F τ net, x net, y net, z F F τ ext, x ext, y ext, z /13/17

14 Center o Gravity q The torque due to the gravitational orce on an object o mass M is the orce Mg acting at the center o gravity o the object q I g is uniorm over the object, then the center o gravity o the object coincides with its center o mass q I the object is homogeneous and symmetrical, the center o gravity coincides with its geometric center /13/17

15 Where is the Center o Mass? q Assume m 1 1 kg, m 3 kg, and x 1 1 m, x 5 m, where is the center o mass o these two objects? m x + m A) x CM 1 m B) x CM m C) x CM 3 m D) x CM 4 m E) x CM 5 m x 1 1 CM m + 1 /13/17 m x

16 Center o Mass (CM) q An object can be divided into many small particles n Each particle will have a speciic mass and speciic coordinates q The x coordinate o the center o mass will be x CM i i mx q Similar expressions can be ound or the y coordinates i m i i /13/17

17 Center o Gravity (CG) q All the various gravitational orces acting on all the various mass elements are equivalent to a single gravitational orce acting through a single point called the center o gravity (CG) Mg CG m g 1 1 x x CG 1 ( m1 + m + m3 +!) g + m g x + m g x +! CG x CG q I g g 3 1 g! q then x CG m1x1 + mx + m m + m + m 1 3 x3 +! +! 3 mi x m i i /13/17

18 CG o a Ladder q A uniorm ladder o length l rests against a smooth, vertical wall. When you calculate the torque due to the gravitational orce, you have to ind center o gravity o the ladder. The center o gravity should be located at A B D C E /13/17

19 Ladder Example q A uniorm ladder o length l rests against a smooth, vertical wall. The mass o the ladder is m, and the coeicient o static riction between the ladder and the ground is µ s Find the minimum angle θ at which the ladder does not slip. /13/17

20 Problem-Solving Strategy 1 q Draw sketch, decide what is in or out the system q Draw a ree body diagram (FBD) q Show and label all external orces acting on the object q Indicate the locations o all the orces q Establish a convenient coordinate system q Find the components o the orces along the two axes q Apply the irst condition or equilibrium q Be careul o signs F F net, x net, y F F ext, x ext, y 0 0 /13/17

21 Which ree-body diagram is correct? q A uniorm ladder o length l rests against a smooth, vertical wall. The mass o the ladder is m, and the coeicient o static riction between the ladder and the ground is µ s gravity: blue, riction: orange, normal: green A B C D /13/17

22 q A uniorm ladder o length l rests against a smooth, vertical wall. The mass o the ladder is m, and the coeicient o static riction between the ladder and the ground is µ s Find the minimum angle θ at which the ladder does not slip. F F x y P x n mg P x P 0 n mg 0 x,max µ n µ mg s s mg /13/17

23 Problem-Solving Strategy q Choose a convenient axis or calculating the net torque on the object n Remember the choice o the axis is arbitrary q Choose an origin that simpliies the calculations as much as possible n A orce that acts along a line passing through the origin produces a zero torque q Be careul o sign with respect to rotational axis n n n positive i orce tends to rotate object in CCW negative i orce tends to rotate object in CW zero i orce is on the rotational axis q Apply the second condition or equilibrium τ net, z τ ext, z 0 /13/17

24 Choose an origin O that simpliies the calculations as much as possible? q A uniorm ladder o length l rests against a smooth, vertical wall. The mass o the ladder is m, and the coeicient o static riction between the ladder and the ground is µ s Find the minimum angle. A) B) C) O D) O O mg mg mg mg O /13/17

25 q A uniorm ladder o length l rests against a smooth, vertical wall. The mass o the ladder is m, and the coeicient o static riction between the ladder and the ground is µ s Find the minimum angle θ at which the ladder does not slip. τ sinθ cosθ θ min O τ min min n tan + τ Pl sinθ tanθ 1 + τ min min 1 ( ) µ s g + τ P l mg cosθ min mg mg P µ mg tan 1 s 1 [ ] (0.4) 0 1 µ! 51 s mg /13/17

26 Problem-Solving Strategy 3 q The two conditions o equilibrium will give a system o equations q Solve the equations simultaneously q Make sure your results are consistent with your ree body diagram q I the solution gives a negative or a orce, it is in the opposite direction to what you drew in the ree body diagram q Check your results to conirm F F τ net, x net, y net, z F F τ ext, x ext, y ext, z /13/

27 Horizontal Beam Example q A uniorm horizontal beam with a length o l 8.00 m and a weight o W b 00 N is attached to a wall by a pin connection. Its ar end is supported by a cable that makes an angle o φ 53 with the beam. A person o weight W p 600 N stands a distance d.00 m rom the wall. Find the tension in the cable as well as the magnitude and direction o the orce exerted by the wall on the beam. /13/17

28 Horizontal Beam Example q The beam is uniorm n So the center o gravity is at the geometric center o the beam q The person is standing on the beam q What are the tension in the cable and the orce exerted by the wall on the beam? /13/17

29 Horizontal Beam Example, q Analyze n Draw a ree body diagram n Use the pivot in the problem (at the wall) as the pivot n This will generally be easiest n Note there are three unknowns (T, R, θ) /13/17

30 Horizontal Beam Example, 3 q The orces can be resolved into components in the ree body diagram q Apply the two conditions o equilibrium to obtain three equations q Solve or the unknowns /13/17

31 /13/17 Horizontal Beam Example, 3 0 sin sin 0 cos cos + b p y x W W T R F T R F φ θ φ θ N m m N m N l l W W d T l W W d l T b p b p z 313 )sin53 (8 ) )(4 (00 ) )( (600 sin ) ( 0 ) ( ) )( sin ( + +! φ φ τ N N T R T T W W T T W W R R b p b p 581 cos 71.7 )cos53 (313 cos cos 71.7 sin sin tan sin sin tan cos sin 1 + +!!! θ φ φ φ θ φ φ θ θ θ

32 Rotational Kinetic Energy q An object rotating about z axis with an angular speed, ω, has rotational kinetic energy q The total rotational kinetic energy o the rigid object is the sum o the energies o all its particles 1 K K mr ω R i i i i i 1 1 KR mr ω Iω i i i q Where I is called the moment o inertia q Unit o rotational kinetic energy is Joule (J) February 13, 017

33 Work-Energy Theorem or pure Translational motion q The work-energy theorem tells us q Kinetic energy is or point mass only, ignoring rotation. q Work W net ΔKE Wnet KE KE F 1 dw d s i mv 1 mv i q Power dw d s P F F v dt dt February 13, 017

34 Mechanical Energy Conservation q Energy conservation q When W nc 0, Wnc Δ K+ΔU K + U Ui + Ki q The total mechanical energy is conserved and remains the same at all times 1 1 mv i + mgyi mv + q Remember, this is or conservative orces, no dissipative orces such as riction can be present mgy February 13, 017

35 Total Energy o a System q A ball is rolling down a ramp q Described by three types o energy n Gravitational potential energy U n Translational kinetic energy Mgh n Rotational kinetic energy 1 Mv 1 Iω 1 1 E MvCM + Mgh+ Iω q Total energy o a system K t Kr CM February 13, 017

36 Work done by a pure rotation q Apply orce F to mass at point r, causing rotation-only about axis q Find the work done by F applied to the object at P as it rotates through an ininitesimal distance ds o dw F d s F cos(90 ϕ) ds F sinϕds Fr sinϕdθ q Only transverse component o F does work the same component that contributes to torque dw τdθ February 13, 017

37 February 13, 017 Work-Kinetic Theorem pure rotation q As object rotates rom θ i to θ, work done by the torque q I is constant or rigid object q Power τω θ τ dt d dt dw P i i i i i d I d dt d I d I d dw W θ θ θ θ θ θ θ θ θ θ ω ω θ ω θ α θ τ 1 1 i I I d I d I W i i ω ω ω ω ω ω θ θ θ θ

38 q Ex: An motor attached to a grindstone exerts a constant torque o 10 N-m. The moment o inertia o the grindstone is I kg-m. The system starts rom rest. n n n Find the kinetic energy ater 8 s 1 τ K Iω 1600J ω ω + αt 40 rad/s α 5 rad/s I i Find the work done by the motor during this time W θ τdθ τ ( θ θ ) J i θi 1 ( θ θ ) ωt+ 160rad αt 1 1 i i W K Ki Iω Iωi 1600 J Find the average power delivered by the motor n P avg dw watts dt 8 Find the instantaneous power at t 8 s P τω watts February 13, 017

39 Work-Energy Theorem q For pure translation 1 1 W Δ K K K mv mv net cm cm, cm, i i q For pure rotation 1 1 Wnet Δ Krot Krot, Krot, i Iω Iωi q Rolling: pure rotation + pure translation W Δ K ( K + K ) ( K + K ) net total rot, cm, rot, i cm, i Iω + mv Iωi + mvi February 13, 017

40 Energy Conservation q Energy conservation q When W nc 0, W Δ K +ΔU nc K + K + U K + K + U rot, cm, rot, i cm, i i q The total mechanical energy is conserved and remains the same at all times I ωi + mvi + mgyi Iω + mv + mgy q Remember, this is or conservative orces, no dissipative orces such as riction can be present total February 13, 017

41 Total Energy o a Rolling System q A ball is rolling down a ramp q Described by three types o energy n Gravitational potential energy U Mgh n Translational kinetic energy 1 Kt Mv n Rotational kinetic energy 1 Kr Iω 1 E Mv + Mgh + 1 Iω q Total energy o a system February 13, 017

42 A Ball Rolling Down an Incline q A ball o mass M and radius R starts rom rest at a height o h and rolls down a 30 slope, what is the linear speed o the ball when it leaves the incline? Assume that the ball rolls without slipping mvi + mgyi + Iω i mv + mgy + Iω Mgh + 0 Mv Iω 0 I MR Mgh Mv + MR Mv + ω 5 v R v R 1 5 Mv 10 v ( gh) 7 February 13, 017 1/

43 Rotational Work and Energy q A ball rolls without slipping down incline A, starting rom rest. At the same time, a box starts rom rest and slides down incline B, which is identical to incline A except that it is rictionless. Which arrives at the bottom irst? q q Ball rolling: 1 mvi + mgy Box sliding i mgh mv + Iω 1 1 ω i mv + mgy + Iω mv + mr ( v / R) mv 5 10 I 1 1 mv i + mgyi mv + mgy 1 7 sliding: mgh mv rolling: mgh mv 10 February 13, 017

44 Blocks and Pulley q Two blocks having dierent masses m 1 and m are connected by a string passing over a pulley. The pulley has a radius R and moment o inertia I about its axis o rotation. The string does not slip on the pulley, and the system is released rom rest. q Find the translational speeds o the blocks ater block descends through a distance h. q Find the angular speed o the pulley at that time. February 13, 017

45 q Find the translational speeds o the blocks ater block descends through a distance h. K + K + U K + K + U rot, cm, rot, i cm, i i m1 v + mv + Iω ) + ( m1 gh m gh) ( 1 I ( m1 + m + ) v mgh m1 gh R v ( m m1 ) gh m1 + m + I / R 1/ q Find the angular speed o the pulley at that time. ω v R 1 R ( m m1 ) gh m1 + m + I / R 1/ February 13, 017

46 Angular Momentum q Same basic techniques that were used in linear motion can be applied to rotational motion. n F becomes τ n m becomes I n a becomes α n v becomes ω n x becomes θ q Linear momentum deined as p mv q What i mass o center o object is not moving, but it is rotating? q Angular momentum L Iω February 13, 017

47 Angular Momentum I q Angular momentum o a rotating rigid object n L has the same direction as ω * n L is positive when object rotates in CCW n L is negative when object rotates in CW q Angular momentum SI unit: kg-m /s Calculate L o a 10 kg disk when ω 30 rad/s, R 9 cm 0.09 m L Iω and I MR / or disk I L ω ω! L! L 1/MR ω ½(10)(0.09) (30) 1.96 kgm /s * When rotation is about a principal axis February 13, 017

48 Angular Momentum II q Angular momentum o a particle L Iω mr ω mv r mvr sinφ q Angular momentum o a particle rpsinφ L r pm( r v) n r is the particle s instantaneous position vector n p is its instantaneous linear momentum n Only tangential momentum component contribute n Mentally place r and p tail to tail orm a plane, L is perpendicular to this plane February 13, 017

49 Angular Momentum o a Particle in Uniorm Circular Motion Example: A particle moves in the xy plane in a circular path o radius r. Find the magnitude and direction o its angular momentum relative to an axis through O when its velocity is v. q The angular momentum vector points out o the diagram q The magnitude is L rp sinθ mvr sin(90 o ) mvr q A particle in uniorm circular motion has a constant angular momentum about an axis through the center o its path O February 13, 017

50 Angular Momentum and Torque q Rotational motion: apply torque to a rigid body q The torque causes the angular momentum to change q The net torque acting on a body is the time rate o change o its angular momentum d Fnet Σ F p τnet dt d Σ τ L dt q Στ and L are to be measured about the same origin q The origin must not be accelerating (must be an inertial rame) February 13, 017

51 Ex4: A Non-isolated System A sphere mass m 1 and a block o mass m are connected by a light cord that passes over a pulley. The radius o the pulley is R, and the mass o the thin rim is M. The spokes o the pulley have negligible mass. The block slides on a rictionless, horizontal surace. Find an expression or the linear acceleration o the two objects. a! τ m ext 1 a! gr February 13, 017

52 Masses are connected by a light cord. Find the linear acceleration a. Use angular momentum approach No riction between m and table Treat block, pulley and sphere as a nonisolated system rotating about pulley axis. a! As sphere alls, pulley rotates, block slides Constraints: Equal v's and a's or block and sphere v ωr or pulley α dω / dt a αr dv / dt Ignore internal orces, consider external orces only Net external torque on system: Angular momentum o system: (not constant) dl dt sys τ net mgr 1 about center o wheel L sys m vr + m vr + Iω m vr + m vr MR ω m1 ar + mar + MR α (m1 R + mr + MR)a τnet m1 g a M + m + m 1 I m gr same result ollowed rom earlier method using 3 FBD s & nd law 1 February 13, 017 a!

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