Physics

Simple Harmonic Motion - 3 Session

Session Objectives

Session Objective Problems

Class Exercise - 1 O is a point of suspension of a simple pendulum of length OA = 1.8 m. N is a nail vertically below O, where ON = 0.9 m. The bob starts from A and returns to A after a complete swing. What is the time taken to complete the swing?

Solution For the journey from A to B, let the time be Similarly, for t BC Total time from A back to A is = 2[ ] = 2.28 s

Class Exercise - 2 Determine the period of small oscillations of a mathematical pendulum, i.e. a ball suspended by a thread = 20 cm in length, if it is located in a liquid whose density is n = 3 times less than that of the ball. The resistance of the liquid is to be neglected.

Solution In the figure, d and  are density of ball and liquid respectively. Then in equilibrium, If A is the area and x is the displacement of the liquid, then

Solution contd.. Compare this by equation a =  2 x So this represents the SHM. The time period T is given by Also d = 3 T = 1.1 s

Class Exercise - 3 A block is resting on a piston which is moving vertically with an SHM of period 1 s. At what amplitude of motion will the block and piston separate?

Solution For the block, mg – R = ma or R = m(g – a) In order to separate the block, R = 0 or a = g Now a =  2 x  g =  2 x = m

Class Exercise - 4 A plank with a body of mass m placed on it starts moving straight up according to the law X = a(1 – cost), where X is the displacement from the initial position,  = 11 s –1. Find the time dependence of the force that the body exerts on the plank. If a = 4.0 cm plot this dependence.

Solution Plot is shown in the figure.

Class Exercise - 5 Find the time period of a pendulum of infinite length. Assume bob to be near to surface of earth.

Solution Restoring force on the bob = mg sin( + ) Force on the bob = –mg{sin( + )} If  is small,  is also small. Hence,

Solution contd..

Class Exercise - 6 In the arrangement shown in the figure, the particle m 1 rotates in a radius r on a smooth horizontal surface with angular velocity  0. If m 2 is displaced slightly in the vertical direction, find the time period of oscillation.

Solution Considering the equilibrium of m 2, we have Now let mass m 2 is displaced downward by a distance x. Then radius of circular path of m 1 decreases to (r – x). Applying the conservation of angular momentum, we have

Solution contd.. Tension is also increased as shown below. As a result, m 2 gets a restoring force given by

Solution contd..

Class Exercise - 7 Find the frequency of small oscillations of a thin uniform vertical rod of mass m and length is hinged at point O. The combined stiffness of the spring is equal to K. The mass of the spring is negligible.

Solution A displaced position of the rod through an angle  is shown in the given figure. The displacement of spring is x. Let K 1 and K 2 be the stiffness of the springs respectively. Considering the torques acting on the rod, we have

Solution contd.. When  is small, cos  = 1 and sin  =  Solving it for , we get Hence, the motion is SHM.

Class Exercise - 8 In the arrangement shown in the following figure, the sleeve M of mass m = 0.2 kg is fixed between two identical spring whose combined force constant K = 20 N/m. The sleeve can slide without friction over a horizontal bar AB. The arrangement rotates with a constant angular velocity  = 4 rad/s about a vertical axis passing through the middle of the bar. Find the period of small oscillations of the sleeve?

Solution We will analyse the problem relative to the rotating bar AB. As the acceleration of bar will be centripetal, a pseudo force will act on the sleeve away from centre and will be of magnitude m  2 x. If the sleeve is displaced by x 1, the net force towards the centre is

Solution contd.. Note: If, there will be no oscillation of the sleeve. It will rush to the point B if it is displaced slightly (for K < m 2 ) or will remain in the displaced position (for K = m 2 )

Class Exercise - 9 The friction coefficient between the two blocks shown in the figure is m and the horizontal plane is smooth. (a) Find the time period of system, and (b) magnitude of frictional force between the blocks.

Solution (a)For small amplitude, the two blocks oscillate together. In this case, (b) The acceleration of the blocks at displacement x from mean position Force on upper block = ma =

Solution contd.. The force is provided by friction of the lower block.  Magnitude of frictional force =

Class Exercise - 10 A liquid of density d is kept in a vertical U-tube of uniform cross section A. If the liquid column is slightly depressed and left, show that the resulting motion of the liquid is SHM and find the period.

Solution Let the liquid be depressed by a height x in the right side of the U- tube. Then the liquid rises above on the left-side by a height x.  Restoring force = Pressure × Area = – (2xdg) × A  The motion is SHM. or K = 2dgA

Thank you