Find the time period of a simple pendulum if the mass of the bob is $\displaystyle m$ and mass of the rod is $\displaystyle M$.
Answer:
Spoiler:
Anyone?
Find the time period of a simple pendulum if the mass of the bob is $\displaystyle m$ and mass of the rod is $\displaystyle M$.
Answer:
Spoiler:
Anyone?
Hello fardeen_genYou need the formula for the period of oscillation of a compound pendulum, which is:
$\displaystyle T = 2\pi\sqrt{\frac{I}{mgh}}$
where $\displaystyle I$ is the moment of inertia about the axis of rotation
$\displaystyle m$ is the mass
$\displaystyle h$ is the distance of the centre of mass from the axis of rotation
The moment of inertia of a rod of mass $\displaystyle M$, length $\displaystyle L$ about one end is $\displaystyle \tfrac13ML^2$, and the moment of inertia of the bob is $\displaystyle mL^2$, so $\displaystyle I= \tfrac13ML^2+mL^2$
Total mass = $\displaystyle M+m$
and, taking moments about the axis of rotation:
$\displaystyle (m+M)h = \tfrac12ML+mL$
$\displaystyle \Rightarrow T =2\pi\sqrt{\frac{\tfrac13ML^2+mL^2}{g(\tfrac12ML+m L)}}$
$\displaystyle =2\pi\sqrt{\frac{(\tfrac13M+m)L}{g(\tfrac12M+m)}}$
(I think you have some brackets missing in your answer!)
Grandad