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Rotational Motion
Angular Acceleration


ar
at 
 
at v
v
w
r
 r
t
t
The angular acceleration is 
In uniform circular motion
there is a constant radial
acceleration.
• ar = v2 / r = rw2

If the angular velocity
changes there is
acceleration tangent to the
circle as well as radially.
The Effect of Torque

A tangential force on a mass creates an acceleration.
• Tangential force: Ft = m at
• Tangential acceleration: at = r

The force is associated with a torque.
• Torque: t = r Ft
r
Ft
m
Rotational Law of Acceleration

The force law can be combined with rotational
motion.
• Torque: t = r Ft = r m at = m r2 

If torque replaces force, and angular acceleration
replaces acceleration, this looks like the law of
acceleration.
t  (mr 2 )  I
Torque and Work

A force does work on an
object acting over a
distance.

A torque does work on an
object rotating through an
angle.
W  t ( q )
F
r
q
Rotational Work-Energy

The net work done by forces on an object equals the
change in kinetic energy.

The net work done by torques on an object equals
the change in rotational kinetic energy.
W  t (q )  Krot  12 Iw 2f  12 Iwi2
Rotational Power

As with translational motion,
power is the rate of work
done.
P
P
W
q
t
 tw
t
t
1
2

• About 28 s / century
• 1 part in 108

The kinetic energy is
changing.

The power dissipation is
large:
Iw 2f  12 Iwi2
t
The earth is slowing due to
the tides.
• About 7 billion hp
Rotation and Translation

A rolling wheel is moving
forward with kinetic energy.

A rolling wheel is rotating
with kinetic energy.

The velocity is measured at
the center of mass.

The axis of rotation is at the
center of mass.
• Krot = ½ I w2
• KCM = ½ m v2
v
w
Rolling Energy



The energy of a rolling wheel
is due to both the translation
and rotation.
The velocity is linked to the
angular velocity.
The effective energy is the
same as a wheel rotating
about a point on its edge.
• Parallel axis theorem
K  K CM  K rot
K  12 mv 2  12 Iw 2
K  12 (mR 2  I )w 2
Energy Conserved

A change in kinetic energy is
due to work done on the
wheel.
• Work is from a force
• Force acts as a torque
v
R

Rolling down an incline the
force is from gravity.
• Pivot at the point of contact

The potential energy is
converted to kinetic energy.
F = mg
q