Question

A $15cm$long, $200g$rod is pivoted at one end. A $20g$ball of clay is stuck on the other end. What is the period if the rod and clay swing as a pendulum?

Short answer

Period$T=0.487sec$

Step-by-step solution

Given information

Pivot length $l=15cm$

mass of pivot $m_1=200g=0.2kg$

mass of ball$m_2=20g=0.02kg$

Explanation

Moment of Inertia of rod about pivot $=\frac{m_1{l}^2}{3}$

Moment of Inertia of clay ball about pivot $=m_2{l}^2$

Combine moment of inertia of pivot:

$$\begin{gathered} I=\left(\frac{m_1}{3}+m_2\right)l^2 \\ =\left(\frac{0.2}{3}+0.02\right){(0.15)}^{2}I=0.00195kg{m}^2 \end{gathered}$$

Combined mass role="math" localid="1650048680672" $M=m_1+m_2=0.2+0.02=0.22kg$

Time period

$$\begin{gathered} T=2\mathrm{\pi }\sqrt{\frac{I}{Mgl}} \\ =2\mathrm{\pi }\sqrt{\frac{0.00195}{0.22\times 9.81\times 0.15}} \\ =0.487\sec \end{gathered}$$