Question

Imagine that a tunnel is bored in the Earth's equatorial plane, going completely through the center of the Earth with both ends at the Equator. A mass of \(5.00 \mathrm{~kg}\) is dropped into the tunnel at one end, as shown in the figure. The tunnel has a radius that is slightly larger than that of the mass. The mass is dropped into the center of the tunnel. Neglect air resistance and friction from the tunnel wall. Does the mass ever touch the wall of the tunnel as it falls? If so, which side does it touch first, north, east, south, or west? (Hint: The angular momentum of the mass is conserved if the only forces acting on it are radial.)

Short answer

If so, which side does it touch first? Answer: Yes, the mass will eventually touch the wall as it falls, due to the Coriolis force acting on it. The side on which the mass touches the wall first cannot be determined without solving for the detailed motion of the mass, but it will be either the east or west side.

Step-by-step solution

Identify the forces on the mass.

Initially, as the mass is dropped into the tunnel, there is gravitational force acting on the mass, drawing it towards the center of the Earth. Note that the gravitational force \(F_G = mg\) is a radial force pointing towards the center of the Earth. Additionally, the mass has an angular velocity due to Earth's rotation. This will give rise to the Coriolis force that will act perpendicular to the mass's radial velocity vector.

Calculate the Coriolis force.

Since the mass starts at rest with respect to the Earth's surface, its initial angular velocity equals Earth's angular velocity, which is: \(\omega_\mathrm{E}=\frac{2\pi\mathrm{rad}}{86400\mathrm{s}}\). The Coriolis force is given by: \(F_C = 2m(\boldsymbol{\omega} \times \boldsymbol{v_r})\). As the mass is dropped with an initial velocity of zero, the initial Coriolis force is also zero.

Analyze the motion.

As the mass falls deeper into the tunnel, the gravitational force continues to pull it towards the Earth's center. The mass will also feel a radial acceleration due to gravity. In addition, The Coriolis force becomes non-zero once the mass acquires velocity. For a point mass moving in the equatorial plane, the Coriolis force will act perpendicular to both the radial velocity vector and Earth's rotation axis. This means that the Coriolis force will be directed in the east-west direction, with its exact direction and magnitude depending on the mass's position within the tunnel.

Conserve the angular momentum.

The given hint suggests that angular momentum is conserved, which implies that the torque on the mass due to gravity and Coriolis force is zero. This can be expressed as: \(\boldsymbol{\tau} = \boldsymbol{r} \times \boldsymbol{F_G} + \boldsymbol{r} \times \boldsymbol{F_C} = \boldsymbol{0}\). We observe that the torque due to the gravitational force is zero as it is a radial force (\(\boldsymbol{r}\) and \(\boldsymbol{F_G}\) are collinear), so the Coriolis force plays a major role in conserving the angular momentum.

Determine the motion of the mass with respect to the wall.

As the mass falls, its radial velocity increases, and so does the Coriolis force acting on it due to Earth's rotation. The Coriolis force will cause the mass to move (deviate) to either east or west side of the tunnel, with the magnitude and direction depending on its instantaneous radial position. However, as the mass never touched the wall at the start, the initial Coriolis force was zero, meaning that the mass has to accelerate in the east-west direction to touch the wall. Therefore, the mass will eventually touch the wall as it falls. However, the exact timing and side on which the mass touches the wall cannot be determined without solving for the detailed motion of the mass.

Answer:

Yes, the mass will eventually touch the wall as it falls, due to the Coriolis force acting on it. The side on which the mass touches the wall first cannot be determined without solving for the detailed motion of the mass but it will be either east or west side.

Key concepts

Gravitational Force

Gravitational force is a fundamental interaction that occurs between all objects with mass. It is the attraction that draws objects toward each other and plays a central role in the motion of bodies in space, including the textbook physics problem at hand.

Understanding this force is crucial for solving problems that involve the motion of objects toward the Earth's center. In our problem, the mass in the tunnel experiences a constant gravitational force that accelerates it towards Earth's core. The gravitational force can be calculated using the formula:
\[F_G = mg\]
where \(m\) is the mass of the object and \(g\) is the acceleration due to gravity. As the object falls, the force does not change direction because it always points radially inward, towards the center of our planet.

Gravitational force is pivotal in determining how objects move under its influence and is often regarded as the only force acting on an object in free-fall if we ignore air resistance. Hence, even though gravity causes the object in the tunnel to accelerate towards Earth's center, it is not responsible for any deviation from its original drop trajectory, making it necessary to consider other forces in the equation.

Angular Momentum Conservation

Conservation of angular momentum is a principle in physics which states that if no external torque acts on a system, the total angular momentum of the system remains constant. This concept is frequently applied to systems in rotation, like a mass moving due to Earth's rotation in our provided exercise.

In the absence of external torques, like friction or air resistance which we are ignoring, the angular momentum of the system we are analyzing can be expressed as:
\[L = I\boldsymbol{\times}\times \omega\]
where \(L\) is the angular momentum, \(I\) is the moment of inertia, and \(\omega\) is the angular velocity. Since the only forces acting on the mass are radial (gravitational force), there is no external torque and angular momentum is conserved.

This conservation means that although the mass will accelerate as it falls towards the center of the Earth, its angular momentum relative to the Earth’s rotation will stay constant. What does change, however, is its moment of inertia—since this is dependent on the mass's radial distance from the Earth's axis of rotation. As this distance decreases, the angular velocity will increase to compensate, preserving angular momentum.

Earth's Rotation Effects

Earth's rotation has several significant effects on the motion of objects on its surface and within, including the phenomenon known as the Coriolis force. This fictitious force results from the rotation of the Earth and affects the motion of objects moving relative to Earth's surface.

For an object moving in the Earth's equatorial plane, like the mass in our tunnel problem, the rotation imparts an initial angular velocity that is the same as that of Earth. As the mass is dropped and starts to fall, the conservation of angular momentum implies that its rotational speed will change depending on its distance to the axis of rotation.

The Coriolis force can be visualized as an apparent force that acts perpendicular to the direction of motion and Earth's axis of rotation. This means that for an object falling inside our hypothetical tunnel, its east-west motion will be influenced by this force, potentially pushing the mass towards one side of the tunnel wall. The Coriolis effect is also the reason why weather systems such as hurricanes spin differently in the Northern and Southern Hemispheres.

Radial Acceleration

Radial acceleration refers to the component of acceleration that points towards or away from the center of a circular path. In the context of the problem, as the mass falls, it experiences radial acceleration due to the gravitational force pulling it toward Earth's center.

The formula for radial acceleration for an object moving in a circle of radius \(r\) with speed \(v\) is:
\[a_r = \frac{v^2}{r}\]
In our case, the speed changes as the mass falls toward the center of Earth, and so the value of radial acceleration is not constant. However, the radial component remains always directed towards the center of Earth. This acceleration contributes to the change in velocity of the mass as it falls, which in turn influences the Coriolis force acting on it. The intricacy involved in the mass's motion within this scenario underscores the complexity of motion under gravity and Earth’s rotation effects.