Chapter 5: Circular Motion; Gravitation

1
Chapter 5 Circular Motion Gravitation
2
Sect. 5-1 Uniform Circular Motion

• Motion of a mass in a circle at constant speed.
• Constant speed
• ? The Magnitude (size) of the velocity vector v
  is constant.
• BUT the DIRECTION of v changes continually!

v v constant
r
r
v ? r
3

• The motion of a mass in a circle at
• Constant Speed.
• 
  Is there an acceleration?
• To answer this, think about
• Newtons 1st Law
• 
  
• 
  Newtons 2nd Law!

v v constant
r
r
4

• A mass moving in circle at Constant Speed.
• Acceleration ? Rate of change of velocity
• a (?v/?t)
• 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!!

5
Centripetal (Radial) Acceleration
Look at the velocity change ?v in the limit that
the time interval ?t becomes infinitesimally
small get
Similar triangles ? (?v/v) (?l/r) As ?t ? 0,
?? ? 0, A? B
As ?t ? 0, ?v ? ? v ?v is in the radial
direction ? a ? aR is radial!
6

• (?v/v) (?l/r) ? ?v (v/r)?l
• Note that the acceleration (radial) is
• aR (?v/?t) (v/r)(?l/?t)
• As ?t ? 0, (?l/?t) ? v
• Magnitude
• Direction Radially inward!
• Centripetal ? Toward the center
• Centripetal acceleration
• Acceleration toward the center.

7

• Typical figure for particle moving in uniform
  circular motion, radius r (speed v constant)
• v Tangent to the circle always!
• a aR Centripetal acceleration.
• Radially inward always!
• ? aR ? v always!!

8
Period Frequency

• 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 ? the number of revolutions per
  second.
• ? T (1/f)

9

• Particle moving in uniform circular motion,
  radius r (speed v constant)
• Circumference
• distance around 2pr
• ? Speed
• v (2pr/T) 2prf
• ? Centripetal acceleration
• aR (v2/r) (4p2r/T2)

10
Example 5-1 Acceleration of a revolving ball
A 150-g ball at the end of a string is revolving
uniformly in a horizontal circle of radius 0.600
m. The ball makes 2.00 revolutions per second.
Calculate its centripetal acceleration.
r
r
11
Example 5-2 Moons Centripetal Acceleration
The Moons nearly circular orbit about the Earth
has a radius of about 384,000 km and a period T
of 27.3 days. Calculate the acceleration of the
Moon toward the Earth.