Name: Experiment: First Law of Simple Pendulum
Uploaded: 2019-07-19
Duration: 5 min 16 s
Description: Do you know how to measure the time period using a simple pendulum? The time period of a pendulum is independent of its amplitude when the amplitude is small...

Question

If a tunnel is dug inside the earth (not necessarily through the centre) and a ball is dropped at one end of it. Show that the ball will execute SHM. Determine the time period of the ball in terms of

G

and density of earth and also in terms of

g

and radius of the earth

Accepted Answer

Let the radius of the earth is $R_{e}$ then the volume of the earth= $\frac{4}{3} {πR}^{3}_{e}$

Let the ball be at distance $r$ from the centre of the earth in the tunnel, then the volume of the inner earth (effective size) $\frac{4}{3} {πr}^{3}$ The mass of the inner earth (effective mass for the ball)

$M = \frac{M_{e}}{\frac{4}{3} {πR}_{e}^{3}} \times \frac{4}{3} {πr}^{3} = \frac{M_{e} r^{3}}{R_{e}^{3}}$

The value of acceleration $g$ at this point on the ball = $g = \frac{GM}{R^{2}} = \frac{{GM}_{e} r^{3}}{{R^{2}}_{e}} \times \frac{1}{r^{2}} = \frac{{GM}_{e} r}{{R^{3}}_{e}}$

This is clearly proportional to the distance and towards the mean position, Hence the ball will execute an SHM with the time period

$T = 2 π \sqrt{\frac{displacement}{acceleration}} = 2 π \sqrt{\frac{r}{\frac{{GM}_{e} r}{{R^{3}}_{e}}}} = 2 π \sqrt{\frac{{R^{3}}_{e}}{{GM}_{e}}} = 2 π \sqrt{\frac{{R^{3}}_{e^{}}}{\frac{4}{3} {πR}^{3}_{e} d}}$

Where $d$ is the density of earth

Now, since the value of ' $g$ at a depth $d is g' = g (1 - \frac{d}{R}) = g \frac{(R - d)}{R} = \frac{gr}{R}$

So the time period $T = 2 π \sqrt{\frac{displacement}{aceleration}} = 2 π \sqrt{\frac{r}{\frac{gr}{R}}} = 2 π \sqrt{\frac{R}{g}}$

If a tunnel is dug inside the earth (not necessarily through the centre) and a ball is dropped at one end of it. Show that the ball will execute SHM. Determine the time period of the ball in terms of $G$ and density of earth and also in terms of $g$ and radius of the earth

Important Questions on Simple Harmonic Motion

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