Suppose a vertical tunnel is dug along the diameter of earth assumed to be a sphere of uniform mass having density ρ. If a body of mass m is thrown in this tunnel, its acceleration at a distance y from the centre is given by

The correct option is 4πGρy/3.

To find the acceleration of a body of mass $m$ inside a vertical tunnel dug along the diameter of the Earth, we can analyze the gravitational force acting on the body at a distance $y$ from the center of the Earth.

Step 1: Understand the Shell Theorem
According to Newton's shell theorem, when a body is located inside a solid sphere of uniform density at a distance $y$ from the center, the net gravitational force exerted on the body by the spherical shell outside of radius $y$ is zero. Therefore, the gravitational force acting on the body is solely due to the mass of the sphere of radius $y$ beneath it.

Step 2: Calculate the mass of the inner sphere
Let $M_y$ be the mass of the Earth's sphere of radius $y$. Since the Earth is assumed to be of uniform density $\rho$, the mass is the volume of this smaller sphere multiplied by its density:

$M_y=\text{Volume}\times \text{Density}=\frac{4}{3}\pi y^3\rho$

Step 3: Calculate the gravitational force on the body
Using Newton's law of universal gravitation, the attractive gravitational force $F$ acting on the body of mass $m$ at a distance $y$ is given by:

$F=\frac{G{M}_{y}m}{y^2}$

Substituting the expression for $M_y$ into the equation:

$F=\frac{G·\left(\frac{4}{3}\pi y^3\rho \right)·m}{y^2}$

Simplifying the terms by canceling $y^2$:

$F=\frac{4}{3}\pi G\rho ym$

Step 4: Determine the acceleration
Using Newton's second law of motion ($F=ma$), the acceleration $a$ of the body is:

$a=\frac{F}{m}$

Substituting the force expression:

$a=\frac{\frac{4}{3}\pi G\rho ym}{m}=\frac{4\pi G\rho y}{3}$

Therefore, the acceleration of the body at a distance $y$ from the center is $\frac{4\pi G\rho y}{3}$.