Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 1 / a

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- Solution a

Solution

Take positive

θ

toward positive

x

from the downward vertical, as in the source diagram. The mass has position and velocity

r = (a_{x} + l sin θ, a_{y} - l cos θ, 0), \dot{r} = (\overset{a}{˙}_{x} + l \dot{θ} cos θ, \overset{a}{˙}_{y} + l \dot{θ} sin θ, 0) .

Thus

T = m ∣ \dot{r} ∣^{2} /2

and

V = m g (a_{y} - l cos θ)

. Expanding the squared speed gives

L = \frac{m}{2} [\overset{a}{˙}_{x}^{2} + \overset{a}{˙}_{y}^{2} + l^{2} \dot{θ}^{2} + 2 l \dot{θ} (\overset{a}{˙}_{x} cos θ + \overset{a}{˙}_{y} sin θ)] - m g (a_{y} - l cos θ) .

The prescribed support coordinates are functions of time, rather than additional dynamical variables. Terms depending only on time do not affect the Euler-Lagrange equation for

θ

, but have been retained here to obtain the inertial-frame kinetic and potential energies directly.

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