Question

(a) What is the period of small-angle oscillations of a simple pendulum with a mass of $\mathbf{0}\mathbf{.}\mathbf{1}\mathit{k}\mathit{g}$at the end of a string of length$\mathbf{1}\mathit{m}\mathbf{?}$(b) What is the period of small-angle oscillations of a meter stick suspended from one end, whose mass is$\mathbf{0}\mathbf{.}\mathbf{1}\mathit{k}\mathit{g}\mathbf{?}$

Short answer

The period of small angle oscillations of a sample pendulum is 2.01 s.

The period of small angle oscillation of a meter stick suspended from one end is 1.64 s.

Step-by-step solution

Define oscillations of a simple pendulum.

A pendulum completes one oscillation when it begins at one extreme position A, goes to the opposite extreme point B, and then returns to A. The time period is the amount of time it takes to complete one oscillation. The oscillation's time period remains constant.

Find the period of small angle oscillations of a simple pendulum.

1. The period of small angle oscillations of a sample pendulum is,

$T=2\pi \sqrt{\frac{l}{g}}$

Here, $l$is length of the pendulum and $g$is acceleration due to gravity.

Substitute $1m$for $l$and $9.8m/s^2$for $g.$

$$\begin{gathered} T=2\pi \sqrt{\frac{1m}{9.8m/s^2}} \\ =2.01s \end{gathered}$$

Therefore, the period of small angle oscillations of a simple pendulum is$2.01s.$

Find the period of small angle oscillation of a meter stick suspended from one end.

b. The period of small angle oscillation of a meter stick suspended from one end is,

$T=2\pi \sqrt{\frac{I}{mgl\text{'}}}$

Here,$l\text{'}$is length of stick from its center to the one end,$m$is mass of the scale and $l$is moment of inertia of the scale relative to one end.

The moment of inertia of the scale is,

$I=\frac{1}{3}m{l}^2$

Substitute $\frac{1}{3}m{l}^2$for $I$and $\frac{l}{2}$for $l\text{'}$in equation $T=2\pi \sqrt{\frac{I}{mgl\text{'}}}$

$$\begin{gathered} T=2\pi \sqrt{\frac{\left(\frac{1}{3}m{l}^2\right)}{mg\left(\frac{l}{2}\right)}} \\ =2\pi \sqrt{\frac{2l}{3g}} \end{gathered}$$

Substitute $1m$for $l$and $9.8m/s^2$for $g.$

$$\begin{gathered} T=2\pi \sqrt{\frac{2\left(1m\right)}{3\left(9.8m/s^2\right)}} \\ =1.64s \end{gathered}$$

Therefore, the period of small angle oscillation of a meter stick suspended from one end is$1.64s.$