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Rolling, Torque, and Angular Momentum

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1 Rolling, Torque, and Angular Momentum
Circular Motion II Rolling, Torque, and Angular Momentum

2 Torque, Energy and Rolling
Torque, moment of inertia Newton 2nd law in rotation Rotational Work Rotational Kinetic Energy Rotational Energy Conservation Rolling Motion of a Rigid Object December 29, 2018

3 Mechanical Energy Conservation Review
When Wnc = 0, The total mechanical energy is conserved and remains the same at all times Remember, this is for conservative forces, no dissipative forces such as friction can be present December 29, 2018

4 Preview: Total Energy of a System
A ball is rolling down a ramp Described by three types of energy Gravitational potential energy Translational kinetic energy New: Rotational kinetic energy Total energy of a system December 29, 2018

5 Energy in Rotational Motion
A rotating rigid body consists of mass in motion, so it has Kinetic Energy Recall: m2 m3 r2 r3 Therefore: Now: Let This is called the Moment of Inertia This is the Rotational Kinetic Energy Equation of a rigid body December 29, 2018

6 Example A C B mA=2.0 kg mB=3.0 kg mC=1.0 kg 3.0 m 5.0 m 4.0 m What is the Moment of Inertia of this object when it is rotated through Point A, perpendicular to the plane of the Diagram? What is the Moment of Inertia about the axis coinciding with axis BC? If the Object rotates about Point A, perpendicular to the plane with angular speed of 5.0 rad/s, what is its Kinetic Energy? December 29, 2018

7 The Moment of Inertia about Point A axis is 84 kgm2
Solution A C B mA=2.0 kg mB=3.0 kg mC=1.0 kg 3.0 m 5.0 m 4.0 m What is the Moment of Inertia of this object when it is rotated through Point A, perpendicular to the plane of the Diagram? Since the Mass at Point A lies on the axis of Rotation, its distance r from the origin is zero, so it contributes nothing to the Moment of Inertia December 29, 2018 The Moment of Inertia about Point A axis is 84 kgm2

8 The Moment of Inertia about BC axis is 18 kgm2
Solution A C B mA=2.0 kg mB=3.0 kg mC=1.0 kg 3.0 m 5.0 m 4.0 m What is the Moment of Inertia about the axis coinciding with axis BC? Since the Masses at Point B and C lies on the axis of Rotation, its distance r from the origin is zero, so they contributes nothing to the Moment of Inertia December 29, 2018 The Moment of Inertia about BC axis is 18 kgm2

9 Solution A C B mA=2.0 kg mB=3.0 kg mC=1.0 kg 3.0 m 5.0 m 4.0 m If the Object rotates about Point A, perpendicular to the plane with angular speed of 5.0 rad/s, what is its Kinetic Energy? Since the Mass at Point A lies on the axis of Rotation, its distance r from the origin is zero, so it contributes nothing to the Moment of Inertia December 29, 2018

10 Definition of Torque First we need definition of torque:
Direction: right hand rule. Torque is calculated with respect to (about) a point (Fulcrum). Changing the point can change the torque’s magnitude and direction Magnitude: A torque is an influence which tends to change the rotational motion of an object. One way to quantify a torque is Torque = Force applied x lever arm The lever arm (r┴) is defined as the perpendicular distance from the axis of rotation to the line of action of the force. Fingers will point along r, ensure they will bend in direction of F, thumb will point along axis of rotation in the direction of the Torque.

11 Torque The units of torque are force times distance, or newton-meters.
A torque of 1 N-m (not Joules) is created by a force of 1 newton acting with a lever arm of 1 meter. Example: You exert a Force on a door of 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)? Via the Right Hand Rule, the direction of the Torque is out of the screen (positive, counterclockwise) December 29, 2018

12 These torques are clockwise about A, so negative
Example A truck is on a bridge. Given that the bridge weighs 80000kg and the distance between spans A and B is 50.0m. The truck, whose mass is kg, is located 15.0m from span A. Determine the support forces at both ends of the bridge (use 10 m/s2 for g). In torque problems with heavy extended objects, just assume that the object’s weight is hanging at the centre of mass Torque at B This positive torque is counter clockwise, because it is pushing upward on the bridge pivoted at A These torques are clockwise about A, so negative

13 Example A truck is on a bridge. Given that the bridge weighs 80000kg and the distance between spans A and B is 50.0m. The truck, whose mass is kg, is located 15.0m from span A. Determine the support forces at both ends of the bridge. Force at B Force at A December 29, 2018

14 Torque and angular Acceleration
Recall: This is Newton’s Second Law for Rotation Only the net torque can cause an angular acceleration December 29, 2018

15 Only tangential momentum component contribute
Angular Momentum I Every rotational quantity that we have encountered is the analog of some quantity in translational motion. The analog of Momentum of a particle is Angular Momentum The vector quantity of Angular Momentum is denoted by: r is the particle’s instantaneous position vector p is its instantaneous linear momentum The Magnitude of Angular Momentum is denoted by: l Is the perpendicular distance from velocity (momentum) of particle to origin December 29, 2018 Only tangential momentum component contribute

16 Angular Momentum II Same basic techniques that were used in linear motion can be applied to rotational motion. F becomes  m becomes I a becomes  v becomes ω x becomes θ Linear momentum defined as What if mass of center of object is not moving, but it is rotating? Angular momentum

17 Angular Momentum II Angular momentum of a rotating rigid object
L has the same direction as  L is positive when object rotates in CCW L is negative when object rotates in CW Angular momentum SI unit: kgm2/s December 29, 2018

18 Example Calculate the Angular Momentum of a 10 kg mass rotating 9 cm about an origin, given that the angular velocity is 320 radians per second December 29, 2018

19 Rotational Inertial of Rigid Object
You do not have to memorize these, as the Moment of Inertial for a Specific Rigid Object will be given to you. December 29, 2018

20 Example Calculate the Angular Momentum of a 10 kg rigid disk rotating 9 cm about its axis, given that the angular velocity is 320 radians per second, and the Moment of Inertial of a disk is 1/2mr2. December 29, 2018

21 What Rolls Down Faster Two solid cylinders roll, without slipping, down an incline, Their initial gravitational potential energy is being converted into two types of kinetic energy: translational KE and rotational KE. One cylinder is 10 times heavier than the other, who wins the race. December 29, 2018

22 Hey, mass and radius play no part in the final velocity
Hey, mass and radius play no part in the final velocity. They both get to the bottom at the same time. December 29, 2018

23 Angular momentum III Angular momentum of a system of particles
angular momenta add as vectors be careful of sign of each angular momentum for this case: Right Hand Rule demonstrate that L1 is out of screen and L2 is into screen December 29, 2018

24 Example Two objects are moving as shown in the figure . What is their total angular momentum about point O? m2 m1 December 29, 2018

25 Angular Momentum Conservation
where i denotes initial state, f is final state L is conserved separately for x, y, z direction For an isolated system consisting of particles, For an isolated system is deformable December 29, 2018

26 Angular Momentum Conservation
Angular Momentum is conserved any time an object, or system of objects experiences no net torque Isolated system: net external torque acting on a system is ZERO no external forces net external force acting on a system is ZERO December 29, 2018

27 Change I by curling up or stretching out - spin rate w must adjust
Controlling spin (w) by changing I (moment of inertia): In the air, tnet = 0 L is constant Change I by curling up or stretching out - spin rate w must adjust

28 Example A skater is spinning (making one revolution every 2.0 seconds) with her arms extended and with 5.0 kg in each hand. She pulls her hands to her stomach. If the original distance the weights are from her axis is 1.0 m and when her arms are tucked in this distance is 0.20 m. If her moment of inertia without the weights is 3.0 kgm2 when arms are stretched out and 2.2 kgm2 when arms are in tight, determine her final angular velocity. December 29, 2018

29 Solution The conservation of angular momentum must hold
The moment of Inertia of the system is: Therefore: Inertial angular velocity of the system is: The Final moment of Inertia is: The conservation of angular momentum must hold Therefore: December 29, 2018

30 Work done by a pure rotation
Apply force F to mass at point r, causing rotation-only about axis Find the work done by F applied to the object at P as it rotates through an infinitesimal distance ds Only transverse component of F does work – the same component that contributes to torque December 29, 2018

31 Work-Kinetic Theorem pure rotation
As object rotates from i to f , work done by the torque I is constant for rigid object Power December 29, 2018

32 Work-Energy Theorem For pure translation For pure rotation
Rolling: pure rotation + pure translation December 29, 2018

33 Example An motor attached to a grindstone exerts a constant torque of 10 Nm. The moment of inertia of the grindstone is I = 2 kgm2. The system starts from rest. Find the kinetic energy after 8 s Find the work done by the motor during this time Find the average power delivered by the motor Find the instantaneous power at t = 8 s December 29, 2018

34 We have I, we need to find ω
Solution An motor attached to a grindstone exerts a constant torque of 10 Nm. The moment of inertia of the grindstone is I = 2 kgm2. The system starts from rest. a) Find the kinetic energy after 8 s We have I, we need to find ω December 29, 2018

35 There are two ways to Solve this problem
Solution An motor attached to a grindstone exerts a constant torque of 10 Nm. The moment of inertia of the grindstone is I = 2 kgm2. The system starts from rest. b) Find the work done by the motor during this time There are two ways to Solve this problem Logic Algebra It was the motor that gave the Grindstone all of the kinetic energy. So, the work done by the motor is 1600J

36 Solution An motor attached to a grindstone exerts a constant torque of 10 Nm. The moment of inertia of the grindstone is I = 2 kgm2. The system starts from rest. c) Find the average power delivered by the motor d) Find the instantaneous power at t = 8 s December 29, 2018

37 Energy Conservation (Deeper Understanding)
When Wnc = 0, The total mechanical energy is conserved and remains the same at all times Remember, this is for conservative forces, no dissipative forces such as friction can be present December 29, 2018

38 Total Energy of a Rolling System
A ball is rolling down a ramp Described by three types of energy Gravitational potential energy Translational kinetic energy Rotational kinetic energy Total energy of a system December 29, 2018

39 A Ball Rolling Down an Incline Example
A ball of mass M and radius R starts from rest at a height of h and rolls down a 30 slope, what is the linear speed of the ball when it leaves the incline? Assume that the ball rolls without slipping.

40 Rotational Work and Energy Understanding
A ball rolls without slipping down incline A, starting from rest. At the same time, a box starts from rest and slides down incline B, which is identical to incline A except that it is frictionless. Which arrives at the bottom first? Ball rolling: Box sliding: Therefore since the box’s final velocity is faster, it will arrive first.

41 Example: A Non-isolated System
A sphere mass m1 and a block of mass m2 are connected by a light cord that passes over a pulley. The radius of the pulley is R, and the mass of the thin rim is M. The spokes of the pulley have negligible mass. The block slides on a frictionless, horizontal surface. Find an expression for the linear acceleration of the two objects. December 29, 2018

42 same result followed from earlier method using 3 FBD’s & 2nd law
Masses are connected by a light cord Find the linear acceleration a. Use angular momentum approach No friction between m2 and table Treat block, pulley and sphere as a non- isolated system rotating about pulley axis. As sphere falls, pulley rotates, block slides Constraints: I Ignore internal forces, consider external forces only Net external torque on system: Angular momentum of system: (not constant) same result followed from earlier method using 3 FBD’s & 2nd law December 29, 2018

43 Example A puck of mass m = 0.5 kg is attached to a taut cord passing through a small hole in a frictionless, horizontal surface. The puck is initially orbiting with speed vi = 2 m/s in a circle of radius ri = 0.2 m. The cord is then slowly pulled from below, decreasing the radius of the circle to r = 0.1 m. What is the puck’s speed at the smaller radius? Find the tension in the cord at the smaller radius. December 29, 2018

44 Solution a) Given and Required:
m = 0.5 kg, vi = 2 m/s, ri = 0.2 m, rf = 0.1 m, vf = ? Isolated system?

45 Solution b) Given and Required:
m = 0.5 kg, vi = 2 m/s, ri = 0.2 m, rf = 0.1 m, vf = 4 m/s December 29, 2018

46 Example: A merry-go-round problem
A 40-kg child running at 4.0 m/s jumps tangentially onto a stationary circular merry-go-round platform whose radius is 2.0 m and whose moment of inertia is 20 kg-m2. There is no friction. Find the angular velocity of the platform after the child has jumped on. December 29, 2018

47 The Merry-Go-Round The moment of inertia of the system = the moment of inertia of the platform plus the moment of inertia of the person. Assume the person can be treated as a particle As the person moves toward the center of the rotating platform the moment of inertia decreases: The angular speed must increase since the angular momentum is constant: December 29, 2018

48 Conservation of angular Momentum
Solution: A merry-go-round problem Conservation of angular Momentum Iplatform = 20 kg.m2 VT = 4.0 m/s mchild = 40 kg r = 2.0 m wplatform = 0 rad/s Therefore: December 29, 2018

49 Example A 2m bar is held horizontally and is used to open a crate. The tip of the bar acts as the fulcrum and is inserted 4 cm between the lid and the crate. If 15 N of force is applied perpendicularly to the end of the bar. What is the force applied to the crate lid? A) 345 N B) 453 N C) 508 N D) 642 N E) 750 N 0.04m 2m 15N December 29, 2018

50 Example A large crane picks up a 60 m steel beam. The steel beam is uniform, but the crane cable is attached 3.0 m off centre of the beam. How much vertical force must be applied on the guide rope to keep the beam level given that the beam has a mass 4.0 kg/m and the guide rope is place on the shorter end of the beam ? A) 261 N B) 453 N C) 508 N D) 642 N E) 735 N 27m 3m 60m The guide rope is 27m from the fulcrum, and the centre of mass is 3m from the fulcrum December 29, 2018

51 Example A 3m long bar is held horizontally with the fulcrum positioned 1 metre from the right end. How much vertical force must be applied to the left end of the bar to support a 800 N block on the right end? A) 345 N B) 453 N C) 500 N D) 348 N E) 400 N 1m 3m December 29, 2018

52 Example Newton’s First Law states that a particle is in equilibrium if the net forces acting on it are zero. When do you consider torques in Newton’s First Law? A) When Forces are unbalanced B) You never consider Torques in Newton’s First Law C) When using objects that have dimensions D) Torques are not forces so don’t include E) Torques always cancel out so don’t include When objects have dimensions, they may have forces that act off-axis which angularly accelerate the object, so must be considered December 29, 2018

53 Example What happens to an object that is experiencing a net torque?
A) Nothing B) The object accelerates linearly C) The velocity of the object increases D) The object accelerates angularily E) The object accelerates both linearly and angularly December 29, 2018

54 Multi-Choice When a cow is moving in a circular motion, what accelerations are NOT possible? A) Zero Acceleration B) Inward Acceleration C) Tangential Acceleration D) Angular Acceleration E) Outward Acceleration All objects moving in a circular motion must experience an acceleration towards the centre. If there is a net torque on the cow, it may also experience angular acceleration. If there is a net force tangent to the circular motion (its hooves), if could also experience tangential acceleration. That leaves the other two out of the picture. December 29, 2018

55 Example Four identical rods of length L experience a force as seen below. Rank the magnitude of the torque about the pivot point on the right end of the rod. F F 2. 1. F/2 F F 2F 3. 60o 4. 1=3=4>2 December 29, 2018

56 Example A ruler of length L rotates with a constant velocity, ω, measured in radians per second. Determine the Period of the ruler’s rotation. A) 1/ω B) Ω C) 2π/ω D) ω/2π E) 2π Period = 1/f, so need to convert angular velocity into angular frequency before inverting. December 29, 2018


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