Questions - Answers

Long Answer Question:    (8 Marks)

1. Show that the motion of a simple pendulum is simple harmonic and hence derive an equation for its time period. What is seconds pendulum?

                                                                
A: i. Let OA represent a simple pendulum suspended from a rigid support 'O'. The length of the pendulum 'l' is the distance between the point of suspension 'O' and the point of oscillation 'A' which is the centre of the bob of mass 'm'.    

ii. The heavy bob is displaced gently from A to B making a small angle 'θ' with the vertical and then released. The pendulum then makes oscillations about the equilibrium position in one plane.

iii. At B the force acting down wards (mg) can be resolved into two perpendicular components, one along the length of the string (mg cosθ) and the other tangential to the path of the motion of the bob (mg sin), as shown in figure.
* The component mg cosθ is balanced by tension in the string.
* The component mg sinθ the only un balanced force, takes the pendulum back to its un displaced position and is responsible for the motion of the simple pendulum.
           The component of the force mg sinθ will impart an acceleration to the bob of the simple pendulum.

                            
[-ve sign because the acceleration is opposite to the displacement]
iv. When θ is very small, sinθ = θ, and equation (1) becomes

                                      a = - gθ → (2)

From figure, the length of the arc AB = x = lθ

               
'g' is constant at a given place and l is constant for a given pendulum. Hence   is constant. Therefore we observe [a  -x] that acceleration of the pendulum is proportional to displacement and is opposite to the direction of displacement, the condition of S.H.M. is thus satisfied and we can say the simple pendulum executes simple harmonic motion.
v. Equation (4) comparing with the general expression for S.H.M.

Second's pendulum:

A simple pendulum whose period is two seconds is called a seconds pendulum.
 

Very Short Answer Questions:  (2 Marks)

1. The bob of a pendulum is made of a hollow brass sphere. What happens to the time period of the pendulum, if the bob is filled with water completely? Why?

A: Remains same. Because centre of mass is same for hollow sphere or completely filled
with water. The length of the pendulum does not change 
 

2. A pendulum clock gives correct time at the equator. Will it gain or lose time if it is taken to the poles? Why?
A: Time period 

 It depends on g value, if we go from equator to pole g value increases therefore time period decreases and hence gains time.