Rotational Kinematics

Position In translational motion,

position is represented by a point, such as x.

In rotational motion, position is represented by an angle, such as , and a radius, r.

x

linear

0 5

x = 3

r

0

/2

/2angular

Displacement

Linear displacement is represented by the vector x.

Angular displacement is represented by , which is not a vector, but behaves like one for small values. The right hand rule determines direction.

x

linear

0 5

x = - 4

0

/2

/2angular

Tangential and angular displacement A particle that rotates

through an angle also translates through a distance s, which is the length of the arc defining its path.

This distance s is related to the angular displacement by the equation s = r

r

s

Speed and velocity

The instantaneous velocity has magnitude vT = ds/dt and is tangent to the circle.

The same particle rotates with an angular velocity = d/dt.

The direction of the angular velocity is given by the right hand rule.

Tangential and angular speeds are related by the equation v = r .

r

s

is outward according to

RHR

vT

vT

Acceleration Tangential acceleration is

given by aT = dvT/dt. This acceleration is parallel

or anti-parallel to the velocity. Angular acceleration of this

particle is given by = d/dt. Angular acceleration is

parallel or anti-parallel to the angular velocity.

Tangential and angular accelerations are related by the equation a = r .

is out of page (+z in this

diagram) according to

RHR

rvT

vT

s

Don’t forget centripetal

acceleration.

Problem: Assume the particle is speeding up.a) What is the direction of the

instantaneous velocity, v?b) What is the direction of the

angular velocity, ?c) What is the direction of the

tangential acceleration, aT?d) What is the direction of the

angular acceleration ?e) What is the direction of the

centripetal acceleration, ac?f) What is the direction of the overall

acceleration, a, of the particle?

What changes if the particle is

slowing down?

First Kinematic Equation

v = vo + at (linear form)Substitute angular velocity for velocity.Substitute angular acceleration for

acceleration.

= o + t (angular form)

Second Kinematic Equation

x = xo + vot + ½ at2 (linear form)Substitute angle for position.Substitute angular velocity for velocity.Substitute angular acceleration for

acceleration.

= o + ot + ½ t2 (angular form)

Third Kinematic Equation

v2 = vo2 + 2a(x - xo)

Substitute angle for position.Substitute angular velocity for velocity.Substitute angular acceleration for

acceleration.

2 = o2 + 2( - o)

Practice problem

The Beatle’s White Album is spinning at 33 1/3 rpm when the power is turned off. If it takes 1/2 minute for the album’s rotation to stop, what is the angular acceleration of the phonograph album?

Rotational Energy

Practice problem

The angular velocity of a flywheel is described by the equation = (8.00 – 2.00 t 2). Determine the angular displacement when the flywheel reverses its direction.

Inertia and Rotational Inertia

In linear motion, inertia is equivalent to mass. Rotating systems have “rotational inertia”. I = mr2 (for a system of particles)

I: rotational inertia (kg m2) m: mass (kg) r: radius of rotation (m)

Solid objects are more complicated; we’ll get to those later. See page 278 for a “cheat sheet”.

Sample Problem

A 2.0-kg mass and a 3.0-kg mass are mounted on opposite ends a 2.0-m long rod of negligible mass. What is the rotational inertia about the center of the rod and about each mass, assuming the axes of rotation are perpendicular to the rod?

Kinetic Energy

Bodies moving in a straight line have translational kinetic energyKtrans = ½ m v2.

Bodies that are rotating have rotational kinetic energyKrot = ½ I 2

It is possible to have both forms at once.Ktot = ½ m v2 + ½ I 2

Practice problem

A 3.0 m long lightweight rod has a 1.0 kg mass attached to one end, and a 1.5 kg mass attached to the other. If the rod is spinning at 20 rpm about its midpoint around an axis that is perpendicular to the rod, what is the resulting rotational kinetic energy? Ignore the mass of the rod.

Rotational Inertia

Rotational Inertia Calculations

I = mr2 (for a system of particles) I = dm r2 (for a solid object) I = Icm + m h2 (parallel axis theorem)

I: rotational inertia about center of massm: mass of bodyh: distance between axis in question and

axis through center of mass

Practice problem

A solid ball of mass 300 grams and diameter 80 cm is thrown at 28 m/s. As it travels through the air, it spins with an angular speed of 110 rad/second. What is its

a) translational kinetic energy?

b) rotational kinetic energy?

c) total kinetic energy?

Practice Problem

Derive the rotational inertia of a long thin rod of length L and mass M about a point 1/3 from one end

a) using integration of I = r2 dm

b) using the parallel axis theorem and the rotational inertia of a rod about the center.

Practice Problem

Derive the rotational inertia of a ring of mass M and radius R about the center using the formula I = r2 dm.

Torque and Angular Acceleration I

Equilibrium

Equilibrium occurs when there is no net force and no net torque on a system. Static equilibrium occurs when nothing in the system

is moving or rotating in your reference frame. Dynamic equilibrium occurs when the system is

translating at constant velocity and/or rotating at constant rotational velocity.

Conditions for equilibrium: = 0 F = 0

Torque

r

F

Hinge (rotates)

Direction of rotation

Torque is the rotational analog of force that causes rotation to begin.

Consider a force F on the beam that is applied a distance r from the hinge on a beam. (Define r as a vector having its tail on the hinge and its head at the point of application of the force.)

A rotation occurs due to the combination of r and F. In this case, the direction is clockwise.

What do you think is the direction of the torque?

Direction of torque is INTO THE SCREEN.

Calculating Torque

The magnitude of the torque is proportional to that of the force and moment arm, and torque is at right angles to plane established by the force and moment arm vectors. What does that sound like?

= r F : torque r: moment arm (from point of rotation to point of

application of force) F: force

Practice ProblemWhat must F be to achieve equilibrium? Assume

there is no friction on the pulley axle.

10 kg

F

3 cm

2 cm

2 kg

Torque and Newton’s 2nd Law

Rewrite F = ma for rotating systemsSubstitute torque for force.Substitute rotational inertia for mass.Substitute angular acceleration for acceleration.

= I : torque I: rotational inertia: angular acceleration

Practice ProblemA 1.0-kg wheel of 25-cm radius is at rest on a fixed axis. A force of 0.45 N is applied tangent to the rim of the wheel for 5 seconds.a) After this time, what is the angular velocity of the wheel?b)Through what angle does the wheel rotate during this 5 second period?

Sample problem Derive an expression for the acceleration of a flat disk of

mass M and radius R that rolls without slipping down a ramp of angle .

Practice problem

Calculate initial angular acceleration of rod of mass M and length L.Calculate initial acceleration of end of rod.

Sample problem

Calculate acceleration.Assume pulley has mass M, radius R, and is a uniform disk.m2

m1

Rotational Dynamics Lab

Work and Power in Rotating Systems

Practice ProblemWhat is the acceleration of this system, and the magnitude of tensions T1 and T2? Assume the surface is frictionless, and pulley has the rotational inertia of a uniform disk.

m1 = 2.0 kg

m2 = 1.5 kg

mpulley = 0.45 kgrpulley = 0.25 m

30o

T1

T2

Work in rotating systems W = F • r (translational systems)

Substitute torque for force Substitute angular displacement for displacement

Wrot = • Wrot : work done in rotation : torque : angular displacement

Remember that different kinds of work change different kinds of energy. Wnet = K Wc = -U Wnc = E

Power in rotating systems

P = dW/dt (in translating or rotating systems) P = F • v (translating systems)

Substitute torque for force. Substitute angular velocity for velocity.

Prot = • (rotating systems) Prot : power expended

: torque : angular velocity

Conservation of Energy

Etot = U + K = Constant(rotating or linear system)For gravitational systems, use the center of

mass of the object for calculating UUse rotational and/or translational kinetic

energy where necessary.

Practice Problem

A rotating flywheel provides power to a machine. The flywheel is originally rotating at of 2,500 rpm. The flywheel is a solid cylinder of mass 1,250 kg and diameter of 0.75 m. If the machine requires an average power of 12 kW, for how long can the flywheel provide power?

Practice ProblemA uniform rod of mass M and length L rotates around a pin through one end. It is released from rest at the horizontal position. What is the angular speed when it reaches the lowest point? What is the linear speed of the lowest point of the rod at this position?

Rolling without Slipping

Rolling without slipping

Total kinetic energy of a body is the sum of the translational and rotational kinetic energies. K = ½ Mvcm

2 + ½ I 2

When a body is rolling without slipping, another equation holds true: vcm = r

Therefore, this equation can be combined with the first one to create the two following equations: K = ½ M vcm

2 + ½ Icm v2/R2

K = ½ m 2R2 + ½ Icm 2

Sample Problem

A solid sphere of mass M and radius R rolls from rest down a ramp of length L and angle . Use Conservation of Energy to find the linear acceleration and the speed at the bottom of the ramp.

Sample Problem A solid sphere of mass M and radius R rolls from rest down a ramp

of length L and angle . Use Rotational Dynamics to find the linear acceleration and the speed at the bottom of the ramp.

Angular Momentum of Particles

Sample Problem

A solid sphere of mass M and radius R rolls from rest down a ramp of length L and angle . Use Conservation of Energy to find the linear acceleration and the speed at the bottom of the ramp.

Sample Problem A solid sphere of mass M and radius R rolls from rest down a ramp

of length L and angle . Use Rotational Dynamics to find the linear acceleration and the speed at the bottom of the ramp.

Practice Problem

A hollow sphere (mass M, radius R) rolls without slipping down a ramp of length L and angle . At the bottom of the ramp

a) what is its translational speed?

b) what is its angular speed?

Angular Momentum

Angular momentum is a quantity that tells us how hard it is to change the rotational motion of a particular spinning body.

Objects with lots of angular momentum are hard to stop spinning, or to turn.

Objects with lots of angular momentum have great orientational stability.

Angular Momentum of a particle

For a single particle with known momentum, the angular momentum can be calculated with this relationship:

L = r p L: angular momentum for a single particler: distance from particle to point of rotationp: linear momentum

Practice Problem

Determine the angular momentum for the revolution of

a) the earth about the sun.

b) the moon about the earth.

Practice Problem

Determine the angular momentum for the 2 kg particle shown

a) about the origin.

b) about x = 2.0.y (m)

x (m)5.0

v = 3.0 m/s-5.0

5.0

Angular Momentum of Solid Objectsand Conservation of Angular Momentum

Angular Momentum - solid object For a solid object, angular momentum is analogous

to linear momentum of a solid object. P = mv (linear momentum)

Replace momentum with angular momentum. Replace mass with rotational inertia. Replace velocity with angular velocity.

L = I (angular momentum) L: angular momentum I: rotational inertia w: angular velocity

Practice Problem

Set up the calculation of the angular momentum for the rotation of the earth on its axis.

Law of Conservation of Angular Momentum

The Law of Conservation of Momentum states that the momentum of a system will not change unless an external force is applied. How would you change this statement to create the Law of Conservation of Angular Momentum?

Angular momentum of a system will not change unless an external torque is applied to the system.

LB = LA (momentum before = momentum after)

Practice Problem

A figure skater is spinning at angular velocity o. He brings his arms and legs closer to his body and reduces his rotational inertia to ½ its original value. What happens to his angular velocity?

Practice ProblemA planet of mass m revolves around a star of mass M in a highly elliptical orbit. At point A, the planet is 3 times farther away from the star than it is at point B. How does the speed v of the planet at point A compare to the speed at point B?

Demonstrations

Bicycle wheel demonstrations Gyroscope demonstrations Top demonstration

Precession

Practice Problem

A 50.0 kg child runs toward a 150-kg merry-go-round of radius 1.5 m, and jumps aboard such that the child’s velocity prior to landing is 3.0 m/s directed tangent to the circumference of the merry-go-round. What will be the angular velocity of the merry-go-round if the child lands right on its edge?

Angular momentum and torque

In translational systems, remember that Newton’s 2nd Law can be written in terms of momentum.

F = dP/dt Substitute force for torque. Substitute angular momentum for momentum.

= dL/dt t: torque L: angular momentum t: time

So how does torque affect angular momentum?

If = dL/dt, then torque changes L with respect to time.

Torque increases angular momentum when the two vectors are parallel.

Torque decreases angular momentum when the two vectors are anti-parallel.

Torque changes the direction of the angular momentum vector in all other situations. This results in what is called the precession of spinning tops.

If torque and angular momentum are parallel…

Consider a disk rotating as shown. In what direction is the angular momentum?

F

rConsider a force applied as shown. In what direction is the torque?

The torque vector is parallel to the angular momentum vector. Since = dL/dt, L will increase with time as the rotation speeds.

L is out is out

If torque and angular momentum are anti-parallel…

Consider a disk rotating as shown. In what direction is the angular momentum?

F

rConsider a force applied as shown. In what direction is the torque?

The torque vector is anti-parallel to the angular momentum vector. Since = dL/dt, L will decrease with time as the rotation slows.

L is in is out

If the torque and angular momentum are not aligned…

For this spinning top, angular momentum and torque interact in a more complex way.

Torque changes the direction of the angular momentum.

This causes the characteristic precession of a spinning top.

L = r Fg

r

Fg

L

= dL/dt

Rotation Review

Practice ProblemA pilot is flying a propeller plane and the propeller appears to be rotating clockwise as the pilot looks at it. The pilot makes a left turn. Does the plane “nose up” or “nose down” as the plane turns left?