(^478) „„„„„ A Textbook of Engineering Mechanics
Example 23.8. A prismatic bar AB of weight W is resting on rough rollers rotating with
equal angular velocity in opposite directions as shown below :
Fig. 23.5.
If the bar is so placed on the rollers that its c.g. is displaced from the middle plane and
released, show that the bar executes a simple harmonic motion. Also find the time period, if the
coefficient of friction is μ and the distance between the two rollers is 2 a.
Solution. Let the weight W of the bar AB acting at its centre of gravity G be at a distance
x from its mid point as M as shown in Fig. 23.6.
Fig. 23.6.
Let R 1 = Vertical reaction of the left roller,
F 1 = Force of friction between the bar and left roller, and
R 2 , F 2 = Corresponding values for the right roller.
Taking moments about the left support and equating the same,
R 2 × 2a = W (a + x)
∴ 2
()
2
Wa x
R
a

• 
  

  =
  Similarly 1 (–)
  2
  Wa x
  R
  a

  
  

  We know that force of friction at the left roller,
  11
  (–)
  2
  Wa x
  FR
  a
  μ
  =μ =
  Similarly 22 ()
  2
  Wa x
  FR
  a
  μ+
  =μ =
  and resultant forces of friction,
  21
  () (–)
  ––
  22
  Wa x Wa x
  FF F
  aa
  μ+μ

  
  

  Wx
  a
  μ