Reeta and Geeta prepared a pendulum. The length of Reeta’s pendulum is 30cm and
that of Geeta’s pendulum is 40cm. Whose time period will be more? Why?
Answers
Answer:
A simple pendulum consists of massless and inelastic thread whose one end is fixed to a rigid support and a small bob of mass m is suspended from the other end of the thread. Let I be the length of the pendulum. When the bob is slightly displaced and released, it oscillates about its equilibrium position. Figure shows the displaced position of the pendulum.
Suppose the thread makes an angle θ with the vertical. The distance of the bob from the equilibrium position A is AB. At B, the weight mg acts vertically downwards. This force is resolved into two components.
(a) The component mg cos θ is balanced by the tension in the thread acting along the length towards the fixed point O.
(b) mg sin θ which is unbalanced, acts A perpendicular to the length of thread. This force tends to restore the bob to the mean position. If the amplitude of oscillation is small, then the path of the bob is a straight line.
∴ F = −mg sin θ …... (1)
If the angular displacement is small sin θ ≈ θ
F = −mg θ …... (2)
But θ = x/l
Thus, F = – mg (x/l)
Comparing this equation with Newton’s second law, F = ma we get,
acceleration a = -gx/l …... (3)
Simple Pendulum
(negative sign indicates that the direction of acceleration is opposite to the displacement) Hence the motion of simple pendulum is SHM.
We know that a = −ω2x
Comparing this with (3)
ω2 = g/l or ω = √(g/l) …... (4)
Time period, T = 2π/ω