Question

A' is a huge planet with uniform mass distribution, mass M, radius R. B is a circular tunnel made in it, concentric with planet. Radius of tunnel is r,$(r = \frac{R}{2})$ cross sectional diameter is d (d << r). C is a small ball of mass m (m <<M) which is moving freely inside the tunnel without friction in a uniform circular motion, take acceleration due to gravity on surface of A as g:

• A. If contact force on C is double the gravitational force, its time period =$2\pi \sqrt{\frac{R}{3g}}$
• B. If contact force on C is double the gravitational force, its time period =$2\pi \sqrt{\frac{R}{g}}$
• C. If contact force is zero, time period T, radius of tunnel r, than $T \propto r$
• D. If contact force is zero, time period T, radius of tunnel r, than T would be independent of r.

Answer 1

Answer: (A)

If contact force on C is double the gravitational force, its time period =$2\pi \sqrt{\frac{R}{3g}}$

Answer 2

The gravitational force on the ball C at a distance r from the center of the planet is given by $F_g = \frac{GMmr}{R^3}$. We are given that the acceleration due to gravity on the surface of the planet is $g = \frac{GM}{R^2}$. So, $GM = gR^2$.
The gravitational force can be written as $F_g = \frac{gR^2 mr}{R^3} = \frac{mgr}{R}$.
In this problem, the radius of the tunnel is $r = \frac{R}{2}$.
So, the gravitational force on the ball is $F_g = \frac{m g (R/2)}{R} = \frac{mg}{2}$. This force is directed towards the center of the planet.

The ball C is moving in a uniform circular motion inside the tunnel of radius r, concentric with the planet. The center of the circular motion is the center of the planet O. The radius of the circular motion is r. For uniform circular motion with speed v, the centripetal force required is $F_c = \frac{mv^2}{r}$, directed towards the center O.

The forces acting on the ball are the gravitational force $F_g$ towards O and the contact force N from the tunnel wall. Since the ball is moving in a circular path of radius r around O, the contact force N must be radial. Let's assume the contact force N is positive when directed inwards.
The net force towards the center O is the sum of the gravitational force and the contact force (with appropriate sign).
Net force towards O = $F_g + N = \frac{mv^2}{r}$.
Here, $F_g$ is always towards O. If N is inwards, it is positive. If N is outwards, it is negative.

Consider the first option: If contact force on C is double the gravitational force, its time period =$2\pi \sqrt{\frac{R}{3g}}$.
Let $N = 2F_g$. For the net force to be towards the center, the contact force must be directed inwards.
Net force towards O = $F_g + N = F_g + 2F_g = 3F_g$.
So, $3F_g = \frac{mv^2}{r}$.
$3 \left(\frac{mg}{2}\right) = \frac{mv^2}{r}$.
$\frac{3mg}{2} = \frac{mv^2}{r}$.
$v^2 = \frac{3gr}{2}$.
The time period of uniform circular motion is $T = \frac{2\pi r}{v}$.
$T^2 = \frac{4\pi^2 r^2}{v^2} = \frac{4\pi^2 r^2}{(3gr/2)} = \frac{8\pi^2 r}{3g}$.
$T = 2\pi \sqrt{\frac{2r}{3g}}$.
We are given $r = \frac{R}{2}$. Substitute this into the expression for T:
$T = 2\pi \sqrt{\frac{2(R/2)}{3g}} = 2\pi \sqrt{\frac{R}{3g}}$.
This matches the first option.

Let's check the other options.
Option 2: If contact force on C is double the gravitational force, its time period =$2\pi \sqrt{\frac{R}{g}}$.
Our calculated time period is $2\pi \sqrt{\frac{R}{3g}}$. So, option 2 is incorrect.

Option 3: If contact force is zero, time period T, radius of tunnel r, than $T \propto r$.
If contact force N = 0, then the net force towards O is $F_g$.
$F_g = \frac{mv^2}{r}$.
$\frac{mg}{2} = \frac{mv^2}{r}$.
$v^2 = \frac{gr}{2}$.
The time period $T = \frac{2\pi r}{v} = \frac{2\pi r}{\sqrt{gr/2}} = 2\pi r \sqrt{\frac{2}{gr}} = 2\pi \sqrt{\frac{4\pi^2 r^2 \cdot 2}{gr}} = 2\pi \sqrt{\frac{2r}{g}}$.
So, $T = 2\pi \sqrt{\frac{2r}{g}}$.
This shows that $T \propto \sqrt{r}$. Option 3 states $T \propto r$, which is incorrect.

Option 4: If contact force is zero, time period T, radius of tunnel r, than T would be independent of r.
From the calculation in option 3, $T = 2\pi \sqrt{\frac{2r}{g}}$, which clearly depends on r. So, option 4 is incorrect.

Therefore, only the first option is correct.