Chapter 10 – Dynamics of Rotational Motion

I.Torque – “twisting force”

A.Definition:Torque = Force x Lever Arm

Units:

B.Sign of Torque according to direction of rotation

C.Vector definition of Torque:

with direction given by the “right-hand rule.” The direction of represents:

- the direction of the axis around which the torque is acting, and

- the direction in which the torque is causing or trying to cause a rotation - “right-hand rule.”

II.Torque(s) acting on a rigid body able to rotate around a fixed axis

A.Dynamic Equations:

, used for rotating objects

, used for objects traveling in a straight line

B.Examples:

1.A string is wrapped around the rim of a disk of mass 3 kg and radius 50 cm. The disk can rotate around an axis through its center.

a.Find the moment of inertia of the disk.

b.If a steady pull of 10 N is exerted on the string, then find the angular acceleration of the disk.

c.If a 10 N block is attached to the end of the string and released, then find the angular acceleration of the disk and the acceleration of the block.

2.A hanging mass mA = 2 kg is attached by a string the mass mB = 4 kg that is on a smooth horizontal surface. The string passes over a pulley whose moment of inertia is 0.2 kg m2 and radius is 10 cm. Find the linear accelerations of mA and mB, the angular acceleration of the pulley, and the tension in the string.

3.A pulley is constructed like two cylinders glued together. The inner cylinder has a radius of 10 cm and the outer cylinder has a radius of 20 cm. A 2 kg mass attached to a string is wrapped around the inner cylinder in one direction and a 4 kg mass attached to a different string is wrapped around the outer cylinder in the opposite direction. Find the accelerations of the masses and the angular acceleration of the pulley. (Take the total moment of inertia of the cylinder combination to be I = 0.15 kg m2.)

II.Work and Energy of a Rigid Body Rotating around a Fixed Axis

A.By analogy to the work done in linear motion

,

the work done by a torque acting on an object capable of rotating can be written as:

,

where z is the torque acting on the rigid body and  is the angular displacement.

The instantaneous power is written similarly to as:

P = z .

B. The work energy equation for a system that consists of an object that is rotating around a fixed axis is

E1 + Wnon = E2

E = K + U .

In this case K is just the rotational kinetic energy, K = KR = .

C.The work energy equation for a more general system that donsists of an object rotating around a fixed axis and other parts of the system moving along straight lines.

E1 + Wnon = E2 , where

E = K + U .

In this case

K = KT + KR = (translational KE) + (rotational KE)

KT = mv2 = kinetic energy of any objects in the system that are traveling in a straight line

KR = = kinetic energy of any objects in the system that are rotating around a fixed axis

U = potential energy

D.Examples

1.A horizontal uniform beam of mass M and length L is hinged at a wall. The opposite end of the beam is supported by a string attached to the wall.

a.When the string is cut, find the initial angular acceleration of the beam.

b.What is the initial acceleration of the end of the beam?

c.What is the angular velocity of the beam when it hits the wall?

d.What is the velocity of the middle of the beam when it hits the wall?

2.A block of mass mA = 5 kg is attached to a second block of mass mB = 10 kg that is at rest on a horizontal surface with a coefficient of friction  = 0.20. A spring at its unstretched length is attached to a wall and the other end of the spring is attached to the second mass. Take the spring constant to be k = 400 N/m. The pulley has a moment of inertia I = 0.15 kg m2 and a radius R = 10 cm. Find the speed of mass mA after it drops 7 cm.

III.How do we handle the motion of a rotating object when the axis of rotation is moving?

The analysis of the motion of an object undergoing a rotation with the axis of rotation moving can be broken down into two motions:

Total Motion = (motion of the center of mass) + (motion around the center of mass)

translational motionrotational motion

Translational motion: ,”of” the center of mass

Rotational motion:, “around” the center of mass

A.First situation: translational motion is not related to rotational motion. The translational and rotational quantities are independent of each other.

Example: a spinning ball traveling through the air or the rotating earth orbiting the sun

B.Second situation: translational motion is related to rotational motion.

Example: a wheel rolling along the ground without slipping or a cylinder (e.g., yoyo) unwinding from a string

The diagram shows a wheel of radius R rolling without slipping on a surface. Note that the center of mass of the wheel moves a horizontal distance x in the same amount of time it takes point Q to travel a distance s around the circular path to point Q'.

In this situation then, the distance the center of mass moves horizontally is the same as the arc length s. This also means that the horizontal velocity of the center of mass equals the tangential velocity of point Q around the arc. In addition the acceleration of the center of mass is the same as the tangential acceleration of a point on the rim. In summary:

Examples:

1.A solid sphere starts from rest and rolls without slipping a distance d down an inclined plane. The angle of the incline is 37o. What is its velocity of its center of mass at the bottom of the incline?

a. use dynamic equations

b.use the energy equation

IV.Angular Momentum of a Particle

A.Recall that . Is there a quantity analogous to linear momentum for rotation? Yes! Start with

.

Take cross product:

Remember that = torque, and let , and call it the angular momentum of the particle, so

, the rotational analogue to .

Again, = angular momentum. The direction is found by the "right hand rule"

B.Examples of angular momentum calculations:

1.A particle of mass m is traveling with a velocity v in a circular path of radius r in the x-y plane.

2.A particle of mass m is traveling with a velocity v in a straight line parallel to the x-axis and a distance b from it.

II.Angular Momentum of a System of Particles

Example: Find the angular momentum of a dumbbell shaped object. The mass of the weights on the ends is M and the rod holding the weights has a mass M and length L. The system is rotating around the center with an angular velocity 

III.For the system of particles, the relationship between torque and angular momentum is

, or in terms of the components:

IV.Conservation of Angular Momentum

A.Let the z-axis be the axis of rotation of the system.

Suppose z = 0, then . Or, in other words, Lz = constant, and angular momentum around the z-axis remains constant, or in other words, is conserved:

,

in words, the initial angular momentum in the z direction equals the final angular momentum in the z direction when the system is changed.

B.Examples:

1.A block of mass m is traveling in a circular path of radius r1 with and angular velocity 1 on a smooth horizontal table. When the string is shortened to a radius r2

a.find the new angular velocity

b.By how much did the energy change? What is responsible for this change?

2.A small ball of putty of mass m is traveling at a velocity v toward a thin rod of mass M and length L. The direction of travel of the putty is perpendicular to the rod. The rod is hinged at the top end and the putty collides and sticks to the bottom end.

a.What is the angular velocity of the combination after the putty sticks to the rod?

b.Through what angle did the rod rotate?

3.A person sits on a stool that is able to rotate around a vertical axis. The moment of inertia of the person-stool combination is 100 kg m2. The person is holding a spinning bicycle wheel with the axis of the wheel horizontal and the wheel rotating at 5 rev/s. Take the moment of inertia of the bicycle wheel to be Iwheel = 15 kg m2. The person slowly rotates the axis of the wheel until the axis is vertical. What is the angular velocity of the system?

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