Two simple pendulum of slightly different length, are set off oscillating in step is a time of 20s has elasped, during which time the longer pendulum has completed exactly 10 oscillations. Find the length of each pendulum.

Length of longer pendulum = 99.3 cm

Length of shorter pendulum = 82.2 cm

Explanation:

Since the longer pendulum undergoes 10 oscillations in 20 s, its period T = 20 s/10 = 2 s.

From T = 2π√ (l/g), the length of the pendulum. l = T²g/4π²

substituting T = 2s and g = 9.8 m/s² we have

l = T²g/4π²

= (2 s) ² * 9.8 m/s² : 4π²

= 39.2 m : 4π²

= 0.993 m

= 99.3 cm

Now, for the shorter pendulum to be in step with the longer pendulum, it must have completed some oscillations more than the longer pendulum. Let x be the number of oscillations more in t = 20 s. Let n₁ = number of oscillations of longer pendulum and n₂ = number of oscillations of longer pendulum.

So, n₂ = n₁ + x. Also n₁ = t/T₁ and n₂ = t/T₂ where T₂ = period of shorter pendulum.

t/T₂ = t/T₁ + x

1/T₂ = 1/T₁ + x (1)

Also, the T₂ = t/n₂ = t / (n₁ + x) (2)

From (1) T₂ = T₁ / (T₁ + x) (3)

equating (2) and (3) we have

t / (n₁ + x) = T₁ / (T₁ + x)

substituting t = 20 s and n₁ = 10 and T₁ = 2s, we have

20 s / (10 + x) = 2 / (2 + x)

10 / (10 + x) = 1 / (2 + x)

(10 + x) / 10 = (2 + x)

(10 + x) = 10 (2 + x)

10 + x = 20 + 10x

collecting like terms

10x - x = 20 - 10

9x = 10

x = 10/9

x = 1.11

x ≅ 1 oscillation

substituting x into (2)

T₂ = t/n₂ = t / (n₁ + x)

= 20 / (10 + 1)

= 20/11

= 1.82 s

Since length l = T²g/4π²

substituting T = 1.82 s and g = 9.8 m/s² we have

l = T²g/4π²

= (1.82 s) ² * 9.8 m/s² : 4π²

= 32.46 m : 4π²

= 0.822 m

= 82.2 cm