A tunnel is dug along the diameter of the earth (radius R and mass M). There is a particle of mass ‘m’ at the center of the tunnel. Find the minimum velocity given to the particle so that it just reaches to the surface of the earth is-

• A. √(GM/R)
• B. √(GM/(2R))
• C. √(2GM/R)
• D. It will reach with the help of a negligible velocity

To determine the minimum velocity required for a particle of mass 'm' located at the center of a tunnel dug through the Earth to just reach the surface, we can use the principles of gravitational potential energy and kinetic energy.

Understanding the Forces at Play

Inside the Earth, the gravitational force acting on the particle decreases linearly with distance from the center. At the center, the gravitational force is zero, and as the particle moves towards the surface, it experiences a restoring force due to gravity.

Energy Considerations

The total mechanical energy of the particle must be conserved. The gravitational potential energy at the surface of the Earth can be expressed as:

• Potential Energy (PE) = -GMm/R

At the center, the potential energy is zero. To just reach the surface, the kinetic energy (KE) given to the particle must equal the change in potential energy:

• Kinetic Energy (KE) = 1/2 mv²

Setting Up the Equation

Setting the kinetic energy equal to the potential energy change gives:

• 1/2 mv² = GMm/R

By simplifying this equation, we can solve for the velocity:

• v² = 2GM/R
• v = √(2GM/R)

Final Answer

The minimum velocity required for the particle to just reach the surface of the Earth is:

√(2GM/R)