Question

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance R/2 from the earth's centre where R is the radius of the earth. The wall of the tunnel is frictionless. (a) Find the gravitational force exerted by the earth on a particle of mass m placed in the tunnel at a distance x from the centre of the tunnel. (b) Find the component of this force along the tunnel and perpendicular to the tunnel. (c) Find the normal force exerted by the wall on the particle. (d) Find the resultant force on the particle. (e) Show that the motion of the particle in the tunnel is simple harmonic and find the time period.

Answer 1

Answer:

a

Explanation:

Answer 2

Explanation:

The correct answers are calculated as below:

 $M=\frac{4}{3} \pi R^{3} \rho$

$M^{1}=\frac{4}{3} \pi x_{1}^{3} \rho$

$M^{1}=\left(\frac{M}{R^{3}}\right) x_{1}^{3}$

The earth exerts the gravitational force on an element of mass m positioned in the tunnel at a distance x from the tunnel’s centre.

(a) So, the earth exerts a gravitational force on the particle of mass ‘x’ is

$\mathrm{F}=\frac{\mathrm{GMm}}{\mathrm{x}_{1}^{2}}=\frac{\mathrm{GMm}}{\mathrm{R}^{3}} \frac{\mathrm{x}_{1}^{3}}{\mathrm{x}_{1}^{2}}=\frac{\mathrm{GMm}}{\mathrm{R}^{3}} \mathrm{x}_{1}=\frac{\mathrm{GMm}}{\mathrm{R}^{3}} \sqrt{\mathrm{x}^{2}+\left(\frac{\mathrm{R}^{2}}{4}\right)}$

(b) $F_{y}=F \cos \theta=\frac{G M m x_{1}}{R^{3}} \frac{x}{x_{1}}=\frac{G M m x}{R^{3}}$

(c) $F_{x}=\frac{G M m}{2 R^{2}}$

Wall exerts the normal force $\mathrm{N}=\mathrm{Fx}.$

(d) Resultant force $=\frac{\mathrm{GMmx}}{\mathrm{R} 3}$

(e) Acceleration = Driving force/mass = $\frac{\mathrm{GMmx}}{\mathrm{R} 3}$

$So, a \alpha \times(\text { The body makes } S H M)$

$\therefore \frac{a}{x}=w^{2}=\frac{G M}{R^{3}} \Rightarrow w=\sqrt{\frac{G M}{R^{3}}} \Rightarrow T=2 \pi \sqrt{\frac{R^{3}}{G M}}$

Therefore, SHM is made by the body.