Question

A tunnel is dug along a chord of the earth at a perpendicular distance R/2 from the earth's centre. The wall of the tunnel may be assumed to be frictionless. Find the force exerted by the wall on a particle of mass m when it is at a distance x from the centre of the tunnel.

Answer 1

• Force exerted by the wall on a particle of mass m when it is at a distance x from the center of the tunnel is GMm/2R²

Given -

• Wall of the tunnel is friction less.
• Perpendicular distance = R/2

Let the radius of imaginary sphere is d.

So mass of imaginary sphere is (M') = 4/3 πd³ρ, where ρ is the density of earth

Mass of earth can be given by -

M = 4/3 πR³ρ

By dividing both the equations

M'/M = d³/R³ so M' = d³M/R³

Force between the particle and center of the earth is

F = GmM'/d²

By putting the values-

F = Gmd³M / d² R³

From the diagram we can see that F cos θ component gives the resulting force.

F cos θ = GMmd/R³ Cos θ

By putting the value of cos θ

F = GmM/2R²

Answer 2

Given that,

Perpendicular distance $d= \dfrac{R}{2}$

Let us consider that, a particle at kept on a sphere.

If the mass of imaginary sphere is

$M'=\dfrac{4}{3}\pi d^3\rho$...(I)

Where, d = radius of sphere

The mass of the earth is

$M=\dfrac{4}{3}\pi R^3\times\rho$....(II)

We need to calculate the mass of the sphere

Divided equation (I) by equation (II)

$\dfrac{M'}{M}=\dfrac{\dfrac{4}{3}\pi d^3\rho}{\dfrac{4}{3}\times\pi R^3\rho}$

$\dfrac{M'}{M}=\dfrac{d^3}{R^3}$

$M'=\dfrac{d^3}{R^3}M$

We need to calculate the force exerted by the wall on a particle

Using formula of force

$F=\dfrac{GmM'}{d^2}$

Put the value of M' into the formula

$F=\dfrac{GmMd^3}{d^2R^3}$

$F=\dfrac{GmMd}{R^3}$

According to figure,

$\cos\theta=\dfrac{perpendicular}{hypotanious}$

We need to calculate the net force exerted by the wall on a particle of mass m

Using formula of force

$F_{net}=F\cos\theta$

Put the value into the formula

$F_{net}=\dfrac{GmMd}{R^3}\times\dfrac{R}{2d}$

$F_{net}=\dfrac{GmM}{2R^2}$

Hence, The net force exerted by the wall on a particle of mass m is $\dfrac{GmM}{2R^2}$