The acceleration due to gravity is found up to an accuracy of 4% on a planet. The energy supplied to a simple pendulum of known mass $$m$$ to undertake oscillations of time period $$T$$ is being estimated. If time period is measured to an accuracy of 3%, the accuracy to which $$E$$ is known is _________ %

For a simple pendulum executing small-angle oscillations the well-known time-period formula is

$$T \;=\;2\pi\sqrt{\dfrac{L}{g}},$$

where $$L$$ is the length of the pendulum and $$g$$ is the local acceleration due to gravity.

Because $$L$$ is not given directly, we first express it in terms of the measured quantities $$T$$ and $$g$$. Rearranging the above equation, we obtain

$$L \;=\;\dfrac{gT^{2}}{4\pi^{2}}.$$

Next, we write the expression for the mechanical energy that has to be supplied to the pendulum so that it oscillates with a (small) angular amplitude $$\theta_{0}$$. The total energy in one cycle equals the maximum potential energy at the extreme position:

$$E \;=\;mgh,$$

with $$h$$ being the vertical rise of the bob. For small angles $$h\approx\dfrac{L\theta_{0}^{2}}{2}$$, therefore

$$E \;=\;mg\left(\dfrac{L\theta_{0}^{2}}{2}\right) \;=\;\dfrac{1}{2}\,m\,g\,L\,\theta_{0}^{2}.$$

Substituting $$L=\dfrac{gT^{2}}{4\pi^{2}}$$ from the earlier step, we get

$$E \;=\;\dfrac{1}{2}\,m\,g\left(\dfrac{gT^{2}}{4\pi^{2}}\right)\theta_{0}^{2} \;=\;\dfrac{m\theta_{0}^{2}}{8\pi^{2}}\;g^{2}T^{2}.$$

Thus the energy depends on the measurable quantities as

$$E \;\propto\;g^{2}\,T^{2}.$$

To find the percentage error in $$E$$ we use the rule for propagation of errors in a product of powers: if $$Q \propto a^{p}b^{q}$$, then $$\dfrac{\Delta Q}{Q}\;=\;|p|\dfrac{\Delta a}{a} + |q|\dfrac{\Delta b}{b}.$$

Here $$p=2$$ for $$g$$ and $$q=2$$ for $$T$$. The given percentage uncertainties are

$$\dfrac{\Delta g}{g}=4\% , \qquad \dfrac{\Delta T}{T}=3\%.$$

Therefore,

$$\dfrac{\Delta E}{E} =2\left(\dfrac{\Delta g}{g}\right) +2\left(\dfrac{\Delta T}{T}\right) =2(4\%) + 2(3\%) =8\% + 6\% =14\%. $$

So, the answer is $$14\%$$.