Uniform Circular Motion

1
Uniform Circular Motion
2
Position on a Circle

• Motion in a circle is common.
• The most important measure is the radius (r).
• The position of a point on the circle is
  described by a radial vector .
• Origin is at the center.
• Magnitude is equal everywhere.

r
r
3
Measuring a Circle

• We use degrees to measure position around the
  circle.
• There are 2p radians in the circle.
• This matches 360
• The distance around a circle is s r q, where q
  is in radians.

Dq
q
r
The angular displacement is Dq
4
Period and Frequency

• Movement around a circle takes time.
• The period (T) is the time it takes to complete
  one revolution around the circle.
• The frequency (f) is the number of cycles around
  completed in a time.
• Cycles per second (cps or Hz)
• Revolutions per minute (rpm)
• Frequency is the inverse of period (f 1/T).

5
Cycles or Radians

• Frequency is measured in cycles per second.
• There is one cycle per period.
• Frequency is the inverse of the period, f 1/T.

• Angular velocity is measured in radians per
  second.
• There are 2p radians per period.
• Angular velocity, w 2p/T.
• Angular velocity, w 2pf.

6
Angular Velocity

• Displacement is related to the angle.
• Displacement on the curve (s)
• Angle around the circle (q)
• Velocity has an angular equivalent.
• Linear velocity (v)
• Angular velocity (w)
• Units (rad/s or 1/s s-1)

7
Speed on a Circle

• The circumference of a circle is 2?r.
• The period is T.
• The speed is related to the distance and the
  period or frequency.
• v 2?r/T
• v 2?rf
• v rw

s 2p r
r
8
Velocity on a Circle

• Velocity is a vector change in position compared
  to time.
• As the time gets shorter, the velocity gets
  closer to the tangent.

9
Direction of Motion

• In the limit of very small angular changes the
  velocity vector points along a tangent of the
  circle.
• This is perpendicular to the position.
• For constant w, the magnitude stays the same, but
  the direction always changes.

10
No Slipping

• A wheel can slide, but true rolling occurs
  without slipping.
• As it moves through one rotation it moves forward
  2pR.

w
v 2pR/T wR
v
R
Dx 2pR
11
Point on the Edge

• A point on the edge moves with a speed compared
  to the center, v wr.
• Rolling motion applies the same formula to the
  center of mass velocity, v wR.
• The total velocity of points varies by position.

v 2vCM
vCM
v 0
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