 In translational motion, position is represented by a point, such as x.
 In rotational motion, position is represented by an angle, such as q
, and a radius, r.
p
0 x = 3
5 linear x p
/2 r q
0
3p
/2 angular

 Linear displacement is represented by the vector
D x .
 Angular displacement is represented by
Dq
, which is not a vector, but behaves like one for small values. The right hand rule determines direction.
p
D x = - 4
0 p
/2
5 linear x
Dq
0
3p
/2 angular


 A particle that rotates through an angle
Dq 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
Dq by the equation s = r
Dq.
s
Dq r

 The instantaneous velocity has magnitude v
T
= ds/dt and is tangent to the circle.



The same particle rotates with an angular velocity w
= d q
/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 w
.
v
T s v
T
Dq r w is outward according to
RHR






Tangential acceleration is given by a
T
= dv
T
/dt.
This acceleration is parallel or anti-parallel to the velocity.
Angular acceleration of this particle is given by a
= d w
/dt.
Angular acceleration is parallel or anti-parallel to the angular velocity.
Tangential and angular accelerations are related by the equation a = r a
.
v
T s v
T r w is outward according to
RHR
Don’t forget centripetal acceleration.

f)
a)
What is the direction of the instantaneous velocity, v?
b) c) d) e)
What is the direction of the angular velocity, w
?
What is the direction of the tangential acceleration, a
T
?
What is the direction of the angular acceleration a
?
What is the direction of the centripetal acceleration, a c
?
What is the direction of the overall acceleration, a, of the particle?
What changes if the particle is slowing down?

 v = v o
+ at (linear form)
 Substitute angular velocity for velocity.

 Substitute angular acceleration for w acceleration.
= w o
+ a t (angular form)

 x = x o
+ v o t + ½ at 2 (linear form)
 Substitute angle for position.
 Substitute angular velocity for velocity.

 Substitute angular acceleration for acceleration.
q
= q o
+ w o t + ½ a t 2 (angular form)

 v 2 = v o
2 + 2a(x - x o
)
 Substitute angle for position.
 Substitute angular velocity for velocity.

 Substitute angular acceleration for acceleration.
w
2 = w o
2 + 2 a
( q
q 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?

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




In linear motion, inertia is equivalent to mass .
Rotating systems have “rotational inertia”.
I =
 mr 2 (for a system of particles)
 I: rotational inertia (kg m 2 )
 m: mass (kg)

 r: radius of rotation (m)
Solid objects are more complicated; we’ll get to those later. See page 304 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?

 Bodies moving in a straight line have translational kinetic energy
 K trans
= ½ m v 2 .
 Bodies that are rotating have rotational kinetic energy
 K rot
= ½ I w
2
 It is possible to have both forms at once.
 K tot
= ½ m v 2 + ½ I w
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.



I =
 mr 2 (for a system of particles)
I =
 dm r 2 (for a solid object)
 I = I cm
+ m h 2 (parallel axis theorem)
 I: rotational inertia about center of mass
 m: mass of body
 h: 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) b) translational kinetic energy?
rotational kinetic energy?
c) total kinetic energy?

Practice Problem
Derive the rotational inertia of a long thin rod of length L and mass M a) about a point 1/3 from one end using integration of I =
 r 2 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 =
 r 2 dm.

 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:


t
= 0

F = 0

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?
Hinge (rotates) r
Direction of rotation
Direction of torque is
INTO THE SCREEN.
F



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?
t
= r

F
 t
: torque
 r : moment arm (from point of rotation to point of application of force)
 F : force

What must F be to achieve equilibrium? Assume there is no friction on the pulley axle.
F
3 cm
2 cm
10 kg
2 kg

nd
 Rewrite

F = m a for rotating systems
 Substitute torque for force.
 Substitute rotational inertia for mass.

 Substitute angular acceleration for acceleration.
t
= I a
 t
: torque
 I: rotational inertia
 a
: angular acceleration

Practice Problem
A 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 q
.

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

m
2 m
1
Sample problem
Calculate acceleration.
Assume pulley has mass M, radius R, and is a uniform disk.



A small pulley and a larger attached disk spin together as a hanging weight falls. DataStudio will collect angular displacement and velocity information for the system as the weight falls. The relevant data is:



Diameter of small pulley: 3.0 cm
Mass of small pulley: negligible
Diameter of disk: 9.5 cm
 Mass of disk: 120 g
 Hanging mass: 10 g
See if we can illustrate Newton’s 2 nd
Law in rotational form.

Demonstration calculations

Practice Problem
What is the acceleration of this system, and the magnitude of tensions T
1 and T
Assume the surface is frictionless, and pulley has the rotational inertia of a
2
? uniform disk.
r m pulley pulley
= 0.45 kg
= 0.25 m
T
2
T
1 m
2
= 1.5 kg m
1
= 2.0 kg
30 o



W = F • D r (translational systems)
 Substitute torque for force
 Substitute angular displacement for displacement
W rot
= t • Dq



W rot
: work done in rotation t
: torque
Dq
: angular displacement
 Remember that different kinds of work change different kinds of energy.
 W net
=
D
K W c
= -
D
U W nc
=
D
E



P = dW/dt (in translating or rotating systems)
P = F • v (translating systems)
 Substitute torque for force.
 P
 Substitute angular velocity for velocity.
rot
= t • w
(rotating systems)



P rot
: power expended t
: torque w
: angular velocity

 E tot
= U + K = Constant
 (rotating or linear system)
 For gravitational systems, use the center of mass of the object for calculating U
 Use 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 Problem
A 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?




Total kinetic energy of a body is the sum of the translational and rotational kinetic energies.
 K = ½ Mv cm
2 + ½ I w
2
When a body is rolling without slipping, another equation holds true:
 v cm
= w r
Therefore, this equation can be combined with the first one to create the two following equations:


K = ½ M v cm
2 + ½ I cm
K = ½ m w
2 R 2 + ½ I cm v 2 /R 2 w
2

Sample Problem
A solid sphere of mass M and radius R rolls from rest down a ramp of length L and angle q
. 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 q
. 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 q
. At the bottom of the ramp a) b) what is its translational speed?
what is its angular speed?

 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.

 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 particle
 r: distance from particle to point of rotation
 p: linear momentum

a) b)
Determine the angular momentum for the revolution of the earth about the sun.
the moon about the earth.
Practice Problem

Practice Problem
Determine the angular momentum for the 2 kg particle shown a) about the origin.
y (m) b) about x = 2.0.
5.0
-5.0
v = 3.0 m/s
5.0
x (m)

 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 w
(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.

 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.
 L
B
= L
A
(momentum before = momentum after)

Practice Problem
A figure skater is spinning at angular velocity w 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 Problem
A 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?

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 merrygo-round if the child lands right on its edge?

 In translational systems, remember that
Newton’s 2 nd Law can be written in terms of momentum.


F = d P /dt
 Substitute force for torque.
 Substitute angular momentum for momentum.
t
= d L /dt
 t: torque
 L: angular momentum
 t: time

 If t
= d L /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.

Consider a disk rotating as shown. In what direction is the angular momentum?
Consider a force applied as shown. In what direction is the torque?
The torque vector is parallel to
F the angular momentum vector.
Since t = d L /dt, L will increase with time as the rotation speeds.
r
L t is out is out

Consider a disk rotating as shown. In what direction is the angular momentum?
Consider a force applied as shown. In what direction is the torque?
F
The torque vector is anti-parallel to the angular momentum vector.
Since t = d L /dt, L will decrease with time as the rotation slows.
r
L t is in is out




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.
r L t
= r

F g
 t
F g t = d L /dt
 D
L

Practice Problem
A 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?