1. Uniform Circular Motion

2. Uniform Circular Motion
Motion of a mass in a circle at constant speed.
Constant speed
 Magnitude (size) of velocity vector v is constant.
BUT the DIRECTION of v changes continually!
v = |v| = constant r r
v  r

3. undergoes acceleration!!
Mass moving in circle at constant speed.
Acceleration  Rate of change of velocity
a = (Δv/Δt)
Constant speed  Magnitude (size) of velocity vector v is constant. v = |v| = constant
BUT the DIRECTION of v changes continually!
 An object moving in a circle
undergoes acceleration!!

4. Centripetal (Radial) Acceleration
Motion in a circle from A to point B. Velocity v is tangent to
the circle. a = limΔt 0 (Δv/Δt) = limΔt 0[(v2 - v1)/(Δt)]
As Δt  0, Δv   v & Δv is in the radial direction
 a  ac is radial!

5. a  ac is radial! Similar triangles  (Δv/v) ≈ (Δℓ/r)
As Δt  0, Δθ  0, A B

6. (Δv/v) = (Δℓ/r)  Δv = (v/r)Δℓ Direction: Radially inward!
Note that the acceleration (radial) is
ac = (Δv/Δt) = (v/r)(Δℓ/Δt)
As Δt  0, (Δℓ/Δt)  v and
Magnitude: ac = (v2/r)
Direction: Radially inward!
Centripetal  “Toward the center”
Centripetal acceleration:
Acceleration toward the center.

7.  ac  v always!! ac = (v2/r) Radially inward always!
Typical figure for particle moving in uniform circular motion, radius r (speed v = constant):
v : Tangent to the circle always!
a = ac: Centripetal
acceleration.
Radially inward always!
 ac  v always!!
ac = (v2/r)

8. Period & Frequency  T = (1/f)
A particle moving in uniform circular motion of radius r (speed v = constant)
Description in terms of period T & frequency f:
Period T  time for one revolution (time to go around once), usually in seconds.
Frequency f  number of revolutions per second.
 T = (1/f)

9. Particle moving in uniform circular motion, radius r (speed v = constant)
Circumference
= distance around= 2πr
 Speed:
v = (2πr/T) = 2πrf
 Centripetal acceleration:
ac = (v2/r) = (4π2r/T2)

10. Example 4.6: Centripetal Acceleration of the Earth
Problem 1: Calculate the centripetal acceleration of the
Earth as it moves in its orbit around the Sun.
NOTE: The radius of the Earth's orbit (from a table) is
r ~ 1.5  1011 m
Problem 2: Calculate the centripetal acceleration due to
the rotation of the Earth relative to the poles
& compare with geo-stationary satellites.
NOTE: From a table, the radius of the Earth is r ~ 6.4  106 m
& the height of geo-stationary satellites is ~ 3.6  107 m

11. Tangential & Radial Acceleration
Consider an object moving in a curved path. If the speed |v| of object is changing, there is an acceleration in the direction of motion.
≡ Tangential Acceleration
at ≡ |dv/dt|
But, there is also always a radial (or centripetal) acceleration perpendicular to the direction of motion.
≡ Radial (Centripetal) Acceleration
ar = |ac| ≡ (v2/r)

12. Total Acceleration
 In this case there are always two  vector components of the acceleration:
Tangential: at = |dv/dt| & Radial: ar = (v2/r)
Total Acceleration is the vector sum: a = aR+ at

13. Tangential & Radial acceleration
Example 4.7: Over the Rise
Problem: A car undergoes a constant acceleration of a = 0.3 m/s2 parallel to the roadway. It passes over a rise in the roadway such that the top of the rise is shaped like a circle of radius r = 515 m. At the moment it is at the top of the rise, its velocity vector v is horizontal & has a magnitude of v = 6.0 m/s. What is the direction of the total acceleration vector a for the car at this instant?