1998 Paper 3 Q11

Year: 1998
Paper: 3
Question Number: 11

Course: UFM Mechanics
Section: Simple Harmonic Motion

Difficulty: 1700.0 Banger: 1500.0

SHM simple pendulum large amplitude energy conservation elliptic integral trigonometric substitution binomial expansion approximation period

Problem

Consider a simple pendulum of length $l$ and angular displacement $\theta$, which is {\bf not} assumed to be small. Show that $$ {1\over 2}l \left({\d\theta\over \d t}\right)^2 = g(\cos\theta -\cos\gamma)\,, $$ where $\gamma$ is the maximum value of $\theta$. Show also that the period $P$ is given by $$ P= 2 \sqrt{l\over g} \int_0^\gamma \left( \sin^2(\gamma/2)-\sin^2(\theta/2) \right)^{-{1\over 2}} \,\d\theta \,. $$ By using the substitution $\sin(\theta/2)=\sin(\gamma/2) \sin\phi$, and then finding an approximate expression for the integrand using the binomial expansion, show that for small values of $\gamma$ the period is approximately $$ 2\pi \sqrt{l\over g} \left(1+{\gamma^2\over 16}\right) \,. $$

No solution available for this problem.

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Difficulty Rating: 1700.0

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Banger Rating: 1500.0

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Problem source

Consider a simple pendulum of length $l$ and angular displacement
$\theta$, which is {\bf not} assumed to be small. Show that
$$
{1\over 2}l \left({\d\theta\over \d t}\right)^2 = g(\cos\theta
-\cos\gamma)\,,
$$
where $\gamma$ is the maximum value of $\theta$. Show also that
the period $P$ is given by
$$
P= 2 \sqrt{l\over g} \int_0^\gamma \left(
\sin^2(\gamma/2)-\sin^2(\theta/2)
\right)^{-{1\over 2}} \,\d\theta \,.
$$
By using the substitution $\sin(\theta/2)=\sin(\gamma/2) \sin\phi$,
and then finding an approximate expression for the integrand using
the binomial expansion,
show that for small values of $\gamma$ the period is approximately
$$
2\pi \sqrt{l\over g} \left(1+{\gamma^2\over 16}\right) \,.
$$

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