Past exam of the physics B course of the University of Cambridge / 2021 / Paper 1 / Section B / B7 / b

Let

I

be each sphere's moment of inertia about the fibre, and let

K_{t}

ℓ

exerts restoring torque

- K_{t} ϕ / ℓ

I \ddot{ϕ} = - \frac{K _{t}}{ℓ _{1}} ϕ, ω_{0}^{2} = \frac{K _{t}}{I ℓ _{1}} .

For the two coupled torsional pendulums, the upper fibre has twist

ϕ_{1}

and the joining fibre has relative twist

ϕ_{2} - ϕ_{1}

. The joining fibre exerts equal and opposite torques on the two spheres:

I \ddot{ϕ}_{1} = - \frac{K _{t}}{ℓ _{1}} ϕ_{1} + \frac{K _{t}}{ℓ _{2}} (ϕ_{2} - ϕ_{1}), I \ddot{ϕ}_{2} = - \frac{K _{t}}{ℓ _{2}} (ϕ_{2} - ϕ_{1}) .

Using the measured

ω_{0}

gives

\ddot{ϕ}_{1} = - ω_{0}^{2} [ϕ_{1} - \frac{ℓ _{1}}{ℓ _{2}} (ϕ_{2} - ϕ_{1})], \ddot{ϕ}_{2} = - ω_{0}^{2} \frac{ℓ _{1}}{ℓ _{2}} (ϕ_{2} - ϕ_{1}) .

The fibre's own rotational inertia and damping are neglected, so its twists supply elastic torques without additional dynamic coordinates.

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