Download Rotational Motion

Rotational Motion
Rigid Body

Real objects have mass at
points other than the center
of mass.
ri

Each point in an object can
be measured from an origin
at the center of mass.

If the positions are fixed
compared to the center of
mass it is a rigid body.
Translation and Rotation


The motion of a rigid body
includes the motion of its
center of mass.
This is translational motion
vCM


A rigid body can also move
while its center of mass is
fixed.
This is rotational motion.

Angular Acceleration


ar
at 
 
at v
v

r
 r
t
t
The angular acceleration is 
In uniform circular motion
there is a constant radial
acceleration.
• ar = v2 / r = r2

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
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 2
• KCM = ½ m v2
v

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 I 2
K  12 (mR 2  I ) 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.
F = mg
q
• Pivot at the point of contact

The potential energy is
converted to kinetic energy.
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