Question

Imagine that a tunnel is bored completely through the Earth along its axis of rotation. A ball with a mass of \(5.00 \mathrm{~kg}\) 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 \(C) ?\)

Short answer

Answer 1: The potential energy function is given by PE(x) = G * m * M / x, where G is the gravitational constant, m is the mass of the ball, M is the mass of the Earth, and x is the distance from the center of the Earth. Question 2: What is the speed of the ball when it arrives at the center of the Earth (point C)? Answer 2: To find the speed of the ball, use the conservation of mechanical energy principle and the formula v = √(2 * PE_i / m), where PE_i is the initial potential energy of the ball at the surface of the Earth. Calculate PE_i from step 1 and substitute it into the formula to find the speed of the ball at the center of the Earth.

Step-by-step solution

Calculate the gravitational potential energy of the ball at the surface of the Earth

First, let's calculate the potential energy of the ball at the surface of the Earth. The formula for gravitational potential energy (PE) is given by: PE = G * m * M / r where PE = potential energy G = gravitational constant = \(6.674 \times 10^{-11} \mathrm{N(m/kg)^2}\) m = mass of the ball = \(5.0 \mathrm{kg}\) M = mass of the Earth = \(5.972 \times 10^{24} \mathrm{kg}\) r = distance from the center of the Earth to the surface = \(6.371 \times 10^6 \mathrm{m}\) Calculating the potential energy of the ball at the surface of the Earth: PE = (\(6.674 \times 10^{-11} \mathrm{N(m/kg)^2}\)) * (5.0 \(\mathrm{kg}\)) * (\(5.972 \times 10^{24} \mathrm{kg}\)) / (\(6.371 \times 10^6 \mathrm{m}\))

Determine the potential energy function

We want to find the potential energy (PE) as a function of the distance from the center of the Earth (x). The general formula for gravitational potential energy remains the same: PE(x) = G * m * M / x

Apply conservation of mechanical energy

As the ball falls, the sum of its kinetic and potential energies remains constant. Therefore, Initial Potential Energy + Initial Kinetic Energy = Final Potential Energy + Final Kinetic Energy PE_i + KE_i = PE_f + KE_f At the surface of the Earth, the ball is at rest, so its initial kinetic energy (KE_i) is 0, and at the center of the Earth (point C), its final potential energy (PE_f) is also 0. PE_i + 0 = 0 + KE_f Solve for KE_f: KE_f = PE_i

Calculate final kinetic energy and find the speed of the ball

Now, we can use the kinematic equation for kinetic energy to find the speed of the ball at the center of the Earth: KE = 0.5 * m * v^2 where KE = kinetic energy m = mass of the ball = \(5.0 \mathrm{kg}\) v = speed of the ball We know that KE_f = PE_i, so: 0.5 * m * v^2 = PE_i Now we can solve for the speed of the ball (v): v = \(\sqrt{(2 * PE_i) / m}\) After calculating PE_i in step 1, we can plug it into this formula to find the speed of the ball at the center of the Earth.

Key concepts

Conservation of Mechanical Energy

Understanding the principle of the conservation of mechanical energy is crucial when analyzing the motion of objects under the influence of gravity. This fundamental principle states that when gravity is the only force doing work, the total mechanical energy of a system – the sum of its potential energy and kinetic energy – remains constant.

Let's consider a ball being dropped into a tunnel through the Earth. The ball starts with a certain amount of gravitational potential energy and no kinetic energy, since it's initially at rest. As the ball falls, its potential energy converts to kinetic energy, increasing the ball’s speed. However, the total mechanical energy of the ball, if we neglect air resistance and other non-conservative forces, remains unchanged.

Why does this matter? It allows us to say something very powerful: the potential energy lost by the ball as it falls towards the Earth's center equals the kinetic energy gained at any point in its descent. This relationship guides us in predicting the ball's speed at different locations in the tunnel without knowing the details of the entire journey.

Kinematic Equation

The kinematic equations relate the variables of motion to one another. Each variable represents a value describing the state of motion of an object. In this scenario, we're interested in the ball's velocity as it reaches the center of the Earth. To find this, we use one of the kinematic equations related to energy:

KE = 0.5 * m * v^2

Where KE stands for kinetic energy, m is the mass, and v is the speed or velocity of the object. We know that when the ball reaches the center of the Earth, all of the initial potential energy will have been converted into kinetic energy (assuming conservation of mechanical energy) and the potential energy at the center will be zero.

By equating the initial potential energy to the final kinetic energy, we get the equation:0.5 * m * v^2 = PE_i

Solving for the ball's velocity, v, we'll be able to determine how fast it will be traveling when it gets to the center. This equation is vital because it links the amount of energy an object has due to its position (potential energy) to the energy it has due to its motion (kinetic energy).

Earth's Gravity

The force that pulls the ball towards the center of the Earth is gravity, a fundamental force that acts between two masses. Newton's law of universal gravitation quantifies this force and tells us that every mass attracts every other mass with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Earth's gravity is central to calculating the gravitational potential energy (PE) of the ball in relation to its distance from the Earth's center. The closer the ball gets to the center, the less gravitational potential energy it has, and the higher its speed becomes as more of that energy is converted into kinetic energy.

Gravitational potential energy on Earth can be determined using the formula:PE = G * m * M / r

Where G is the gravitational constant, m is the mass of the object (in this case, the ball), M is the mass of the Earth, and r is the distance from the center of the Earth. The realization that this energy decreases as the ball moves towards the Earth's center is essential when considering the conservation of mechanical energy and predicting the ball’s movement through the tunnel.