Chapter 12
Study Guide for Motion in a Circle
12.1
Uniform circular motion
Skill 12.1 Understand how rotational position and velocity diFer from linear position
and velocity.
So far, the only type of motion we have been dealing with is
translation motion
, which involves no change in
an object’s orientation.
In this chapter, we will look at a diFerent type of motion, called
rotational motion
, which does involve
changes in an object’s orientation (for example, the motion of a spinning record).
De±nition of Rotational Position:
Rotational Position (
ϑ
) is a unitless quantity
measured counterclockwise from the positive xaxis. It increases by
2
π
for each circle
completed by the revolving object. See ±igure 12.5.
De±nition of Rotational Velocity:
Rotational Velocity is the rate at which an object’s
rotational position changes. The direction of rotational velocity is either clockwise or
counterclockwise.
The instantaneous (linear) velocity vector of an object in circular motion is always perpendicular to the object’s
(linear) position vector.
While all points on a rotating object have the same rotational velocity, their linear velocities will diFer.
This makes sense. In order to make one rotation, any point on a record a distance r away from the axis of
rotation will have to travel a distance
2
πr
(the circumference). This means that points further from the axis of
rotation have to travel a greater distance to make one rotation (since
r
is greater), but they still have to do it in
the same amount of time it takes points closer to the axis of rotation. In order to do so, they must have higher
linear speeds.
Skill 12.2 Understand what it means for an object to be in uniform circular motion.
De±nition of Uniform Circular Motion:
In
uniform circular motion
, the
linear
speed
of each point on the object remains constant.
A direct consequence of this is that the rotational position of an object in uniform
circular motion increases linearly with time.
Skill 12.3 Understand why objects in uniform circular motion have a nonzero acceler
ation (called centripetal acceleration).
Since linear velocity is always perpendicular to the position vector, and the position vector is constantly changing,
the direction of the linear velocity vector is also constantly changing.
Since velocity is changing, there must be an acceleration. This acceleration is called
centripetal acceleration
and is directed towards the center of the circular path.
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Forces and uniform circular motion
Skill 12.4 Understand the force responsible for maintaining uniform circular motion.
Any object moving in uniform circular motion is subject to a net force directed towards the center of its circular
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 Spring '08
 Stewart
 Physics, Circular Motion, Moment Of Inertia, Rotation, Uniform Circular Motion, rotational velocity

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