Problem

Imagine that a tunnel is bored completely through the Earth along its axis of rotation. A...

Imagine that a tunnel is bored completely through the Earth along its axis of rotation. A ball with a mass of 5.00 kg is dropped from rest into the tunnel at the North Pole, as shown in the figure. Neglect air resistance and friction from the tunnel wall. Calculate the potential energy of the ball as a function of its distance from the center of the Earth. What is the speed of the ball when it arrives at the center of the Earth (point C)?

Step-by-Step Solution

Solution 1

THINK:

The speed of the ball at the center is found using the principle of the conservation of energy and with ’s gravitational law we find the potential energy of the ball.

SKETCH:

A ball of mass m is dropped into the tunnel at the North Pole as shown in the sketch.

RESERCH:

Mass of the ball is

Density of the earth is

Radius of the earth is

Let the distance of ball from the center of the earth isand the velocity of the ball at the center of the earth is

Initial kinetic energy of the ball

Kinetic energy of the ball at the center of the earth is

Potential energy at the center of the earth is

Universal gravitational constant is

Then the force acting on the ball inside the earth is given by

The potential energy of the ball is

SIMPLIFY:

The change in the potential energy of the ball is

From law of conservation of energy

Velocity of the ball at the center of the earth is

CALCULATE:

ROUND:

We report the result to four significant figures.

The speed of the ball is

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Solutions For Problems in Chapter 12

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