Physics 4A Chapters 11: Rolling, Torque, and Angular Momentum Nothing can bring you peace but yourself. Ralph Waldo Emerson The foolish man seeks happiness in the distance, the wise man grows it under his feet. James Oppenheim Happiness is not a station you arrive at, but a manner of traveling. Margaret B. Runbeck Reading: pages 275 296 Outline: rolling kinetic energy of rolling forces of rolling (read on your own) cross product right-hand rule (RHR) torque as a vector angular momentum Newton s second law in angular form angular momentum of a particle angular momentum of a rigid body conservation of angular momentum gyroscopic motion (if time if not read on your own) Chapter 11: Problem Solving Dynamics problems for a rotating body are based on the vector equation τ = dl / dt. Some problems simply ask for a calculation of a torque or angular momentum using the general definitions. Identify the point to be used as the origin, then evaluate the appropriate vector product. To find a torque, take r to be the vector from the origin to the point of application of the force and evaluate r F. To find the angular momentum of a particle, take r to be the vector from the origin to the particle, then evaluate r p or mr v. You must be given the linear momentum of the particle or else its mass and velocity. To find the angular momentum of a rigid body rotating about a fixed axis, use L z = Iω, where I is the rotational inertia of the body and w is its angular speed. The component of L along the axis of rotation is the most important component and its direction is given by a right-hand rule. Problems involving rolling objects can be solved by considering the forces acting on the object. Many can be also solved using the energy method. In either case, don't forget the condition of no-sliding: the point on the rolling object in contact with the surface is instantaneously at rest with respect to the surface.
A dynamical solution starts with a force diagram. See Section 11-3 of the text for an illustrative example, a wheel rolling down a ramp. Show all forces and label them with algebraic symbols. Since the application points of forces are important for rotation you cannot represent the body as a point. Instead, sketch the body and place the tail of each force vector at the point where the force is applied. Choose a coordinate system to describe the motion of the center of mass, with one axis in the direction of its acceleration, if possible. Sum the force components for each coordinate direction, again taking care with the signs. Equate the sum to the product of the mass and the corresponding component of the acceleration of the center of mass. Then choose a direction for positive rotation. For each force write an expression that gives the torque about the axis of rotation. Be careful about the signs. Sum the torques and equate the sum to Iα. When a wheel accelerates without sliding, a frictional force is present at the point of contact between the body and the surface on which it rolls. Don't forget to include it, even if it is not mentioned in the problem statement. You then write an additional equation relating the acceleration of the center of mass to the angular acceleration about the center of mass: a com = ±αr. The sign depends on the choices made for the directions of positive translation and rotation. If the body is sliding, the magnitude of the force of friction is given by f = μ k N, where μ k is the coefficient of kinetic friction for the body and surface and N is the magnitude of the normal force exerted by the surface on the body. In some cases you can determine the direction of the force before solving the problem and can take it into account in drawing the force diagram and writing Newton's second law. In other cases you must arbitrarily select a direction, then examine the solution to see if the direction of rotation is consistent with your selection. If it is not, rework the problem using the opposite direction. The energy method makes use of either the work-kinetic energy theorem or conservation of mechanical energy. If external forces do work on the object, use W net = ΔK, where W net is the total work done by all forces and ΔK is the change in the kinetic energy. If all forces that do work are internal and conservative, use U i + K i = U f + K f, where U i and U f are the initial and final potential energies and K i and K f are the initial and final kinetic energies. In either case don't forget to include both the kinetic energy of translation and the kinetic energy of rotation. If the object is not sliding, include the auxiliary equation v com = ± ω R. Remember that gravitational potential energy is given by Mgy, where y is the height of the center of mass above the reference height. Notice that the energy method requires fewer equations than the dynamical method. Less information is required and fewer unknowns can be evaluated. Directions of forces and torques enter only insofar as they determine the work or potential energy. The energy method cannot be used to solve for directions except in special cases. Some problems can be solved using the principle of angular momentum conservation. To see if the principle can be used, you must select a system and examine the net external torque acting on it to see if it vanishes. If it does, then angular momentum is conserved. If one component of the total torque vanishes, then that component of the angular momentum is conserved, regardless of whether the other components change.
Some problems deal with two objects that exert torques on each other, thereby changing each other's angular momentum but are subject to no external torques. Write the total angular momentum in terms of the quantities that describe the motions before the objects interact (their initial velocities or angular velocities); write the angular momentum in terms of the quantities that describe the motions after the interaction (their final velocities or angular velocities). Equate the two expressions and solve for the unknown. In some problems one of the objects is moving along a straight line, either before or after the interaction. Don't forget to include its angular momentum. Some problems deal with an object whose rotational inertia changes at some time while it is spinning about a fixed axis. The problem statement gives information that can be used to calculate the initial values I 0 and ω 0, before the rotational inertia changes, and asks for either I or ω at a later time, after they change. If no external torques act, angular momentum is conserved and I 0 ω 0 = Iω. This can be solved for one of the quantities in terms of the others. Mathematical Skills You should know how to find the magnitude and direction of vector products, such as a b. Recall that the magnitude is absinφ, where φ is the smallest angle between a and b when they are drawn with their tails at the same point. You should also remember the right hand rule for finding the direction. The product is perpendicular to the plane of a and b. Curl the fingers of your right hand so they rotate a toward b through the angle φ. Then, the thumb will point in the direction of the vector product. You also need to know how to compute the vector product in terms of the components of the factors: ( a b ) x = a y b z - a z b y (a b ) y = a z b x - a x b z (a b ) z = a x b y - a y b z Question 1 In the figure below, a block slides down a frictionless ramp and a sphere rolls without sliding down a ramp of the same angle θ. The block and sphere have the same mass, start from rest at point A, and descend through point B. (a) In that descent, is the work done by the gravitational force on the block greater than, less than, or the same as the work done by the gravitational force on the sphere? At B, which object has more (b) translational kinetic energy and (c) speed down the ramp?
Question 2 The figure shows three particles of the same mass and the same constant speed moving as indicated by the velocity vectors. Points a, b, c, and d form a square, with point e at the center. Rank the points according to the magnitude of the net angular momentum of the three-particle system when measured about the points, greatest first. Question 3 A bola, which consists of three heavy balls connected to a common point by identical lengths of sturdy string, is readied for launch by holding one of the balls overhead and rotating the wrist, causing the other two balls to rotate in a horizontal circle about the hand. The bola is then released, and its configuration rapidly changes from that in the overhead view of Fig. a (see below) to that of Fig. b. Thus, the rotation is initially around axis 1 through the ball that was held. Then it is around axis 2 through the center of mass. Are (a) the angular momentum and (b) the angular speed around axis 2 greater than, less than, or the same as that around axis 1? Question 4 A rhinoceros beetle rides the rim of a horizontal disk rotating counterclockwise like a merry-goround. If it then walks along the rim in the direction of the rotation, will the magnitudes of the following increase, decrease, or remain the same: (a) the angular momentum of the beetle disk system, (b) the angular momentum and angular velocity of the beetle, and (c) the angular momentum and angular velocity of the disk? (d) What are your answers if the beetle walks in the direction opposite the rotation? Question 5 The figure below gives the angular momentum magnitude L of a wheel versus time t. Rank the four lettered time intervals according to the magnitude of the torque acting on the wheel, greatest first.
Problem 1 (From Exam 2 Spring 2002) A solid steel ball of mass 0.50 kg and diameter 20 cm is held in place against a spring with spring constant k = 100 N/m, compressing the spring a distance x = 30 cm. The ball is then released from rest and rolls without slipping along a horizontal floor. It then makes a smooth transition to an inclined plane and rolls without slipping up the plane as shown in the figure below. x d 30.0 o a) How far up the incline plane (i.e. what is d in the figure) does the ball roll before coming to rest? b) How fast is the ball rolling (i.e. what is the speed of the ball s center of mass) when it has rolled 1.2 m up the inclined plane? Problem 2 A 140 kg hoop rolls along a horizontal floor so that the hoop s center of mass has a speed of 0.150 m/s. How much work must be done on the hoop to stop it?
Problem 3 In the figure below, a solid ball rolls smoothly from rest (starting at height H = 6.0 m) until it leaves the horizontal section at the end of the track at height h = 2.0 m. How far horizontally from point A does the ball hit the floor? Problem 4 Force F = (-8.0 N) î +(6.0 N) ĵ acts on a particle with position vector r = (3.0 m) î + (4.0 m) ĵ. What are (a) the torque on the particle about the origin and (b) the angle between the directions of r and F?
Problem 5 At the instant the displacement of a 2.00 kg object relative to the origin is ˆ ˆ ˆ d = (2.00 m) i + (4.00 m) j (3.00 m) k its velocity is v = (6.00 m/ s) i ˆ + (3.00 m/ s) ˆ j+ (3.00 m/ s) kˆ and it is subject to a force F = (6.00 N) iˆ (8.00 N) ˆj+ (4.00 N) kˆ. Find (a) the acceleration of the object, (b) the angular momentum of the object about the origin, (c) the torque about the origin acting on the object, and (d) the angle between the velocity of the object and the force acting on the object.
Problem 6 A 2.0 kg particle-like object moves in a plane with velocity components v x = 30 m/s and v y = 60 m/s as it passes through the point with (x, y) coordinates of (3.0, -4.0) m. Just then, what is its angular momentum relative to (a) the origin and (b) the point (-2.0, -2.0) m?
Problem 7 The figure below shows a rigid structure consisting of a circular hoop of radius R and mass m, and a square made of four thin bars, each of length R and mass m. The rigid structure rotates at a constant speed about a vertical axis, with a period of rotation of 2.5 s. Assuming R = 0.50 m and m = 2.0 kg, calculate (a) the structure s rotational inertia about the axis of rotation and (b) its angular momentum about that axis. Problem 8 Two disks are mounted (like a merry-go-round) on low-friction bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, with rotational inertia 3.30 kg m 2 about its central axis, is set spinning counterclockwise at 450 rev/min. The second disk, with rotational inertia 6.60 kg m 2 about its central axis, is set counterclockwise spinning at 900 rev/min. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at 900 rev/min, what are their (b) angular speed and (c) direction of rotation after they couple together?
Problem 9 In the figure below, a 1.0 g bullet is fired into a 0.50 kg block attached to the end of a 0.60 m nonuniform rod of mass 0.50 kg. The block-rod-bullet system then rotates about a fixed axis at A. Treat the block as a particle. If the angular speed of the system about A just after impact is 4.75 rad/s, what was the bullet s speed just before impact. Problem 10 In the figure below, a small 50 g block slides down the frictionless surface through height h = 20 cm and then sticks to a uniform rod of mass 100 g and length 40 cm. The rod pivots about point O through the angle θ before momentarily stopping. Find θ.