91影视

• The deviation of restoring torque required for SHM from the actual restoring torque is$1.0\%$.
• Trigonometric functions from Appendix E of the book.

Using the formula for restoring torque required for SHM and actual restoring torque, we can find the angular amplitude for a simple pendulum.

Formula:

Restoring torque required for SHM, $蟿=-L{F}_g\mathrm{蝉颈苍胃}$ (i)

Actual restoring torque is,

$蟿=-L{F}_g胃$ (ii)

So, the deviation of restoring torque required for SHM from the actual restoring torque is given by the difference of equation (i) and (ii) as:

$$\begin{gathered} \frac{\left(\right(-L{F}_g\sin 胃)-(-L{F}_g胃\left)\right)}{-L{F}_g\sin 胃}=0.01 \\ \frac{\left(\right(\sin 胃)-(胃\left)\right)}{\sin 胃}=0.01 \\ 1-\frac{胃}{\sin 胃}=0.01 \end{gathered}$$

We can write for $\mathrm{si}n胃$from Trigonometric Expansions given in APPENDIX E of the book as:

$\sin 胃=胃-\frac{{胃}^3}{6}$

Hence, the angular amplitude of the pendulum is given as:

$$\begin{gathered} 1-\frac{胃}{胃-{\displaystyle \frac{{胃}^3}{6}}}=0.01 \\ 1-\frac{1}{1-{\displaystyle \frac{{胃}^2}{6}}}=0.01 \\ \frac{1-{\displaystyle \frac{{胃}^6}{6}}}{{\displaystyle \frac{{胃}^2}{6}}}=100 \\ \frac{6}{{胃}^2}=101 \\ 胃=\sqrt{\frac{6}{101}} \\ =0.24\mathrm{rad} \\ =14掳 \end{gathered}$$

Therefore, the angular amplitude${胃}_m$ for a simple pendulum is$14掳$.