Q: A simple pendulum is swinging in vertical plane. The ratio of its potential energy when it is making 45° and 90° with the vertical is

(a) 1:1

(b) 1:(√2+1)

(c) √2 ∶ 1

(d) (√2-1) ∶ √2

Ans: (d)

Sol:

P.E when making 45° with vertical

P.E_{1} = m g l(1-cos45) …(i)

P.E when making 90° with vertical

P.E_{2} = m g l …(ii)

$ \displaystyle \frac{P.E_1}{P.E_2} = 1 – cos45 $

$ \displaystyle \frac{P.E_1}{P.E_2} = 1 – \frac{1}{\sqrt{2}} $

$ \displaystyle \frac{P.E_1}{P.E_2} = \frac{\sqrt{2}-1}{\sqrt{2}} $