Motion Of A Ball In Tunnel Through The Earth

Edited By Vishal kumar | Updated on Jul 02, 2025 06:23 PM IST

Imagine a scenario where a tunnel is dug straight through the centre of the Earth, connecting two opposite points on its surface. If a ball is dropped into this tunnel, it will experience a unique and fascinating motion. As it falls, the ball would be pulled by Earth's gravity, accelerating towards the centre. However, once it passes the centre, the gravitational force would start decelerating it, eventually bringing it to a halt on the other side. This oscillatory motion can be likened to a pendulum swinging back and forth or a child on a swing moving to and fro. Just as the child or the pendulum always returns to the centre, the ball in the tunnel would keep oscillating around the Earth's core, illustrating fundamental concepts of physics such as gravitational force, potential energy, and harmonic motion. This thought experiment isn't just a whimsical idea; it connects with real-world concepts like simple harmonic motion and gravity, which play a significant role in everything from how we design roller coasters to how we understand planetary orbits.

Motion Of A Ball In Tunnel Through The Earth

Motion of A Ball In Tunnel Through the Earth

Imagine digging a tunnel straight through the Earth from one side to the other and then dropping a ball into it. As the ball falls, it’s pulled by Earth's gravity towards the centre. At first, it accelerates, moving faster and faster as it gets closer to the core. But as it passes the centre, something interesting happens the gravitational pull starts to slow it down, eventually bringing it to a stop at the other end of the tunnel. Then, it would start falling back again, creating a repetitive back-and-forth motion, similar to the swinging of a pendulum.

Case I:

If the tunnel is along a diameter and the ball is released from the surface. If the ball at any time is at a distance y from the center of the earth as shown in the below figure,

So the restoring force will act on the ball due to gravitation between the ball and the earth.

Acceleration of the particle at the distance y from the centre of the earth is given by

$\begin{aligned} a = \frac{- G M y}{R^{3}}_{and} g = \frac{G M}{R^{2}} \\ So a = \frac{- (g R^{2}) y}{R^{3}} \Rightarrow a = - \frac{g}{R} y \\ Comparing with a = - ω^{2} y \\ ω^{2} = \frac{g}{R} ω = \sqrt{\frac{g}{R}} \Rightarrow T = 2 π \sqrt{(\frac{R}{g})} = 84.6 \min \end{aligned}$

Case II:

If the tunnel is along a chord and the ball is released from the surface. If the ball at any time is at a distance x from the
centre of the tunnel, as shown in the below figure

then the acceleration of the particle at a distance y from the centre of the earth

$a = \frac{- G M y}{R^{3}}$
and using $g = \frac{G M}{R^{2}}_{we get} a = \frac{- (g R^{2}) y}{R^{3}} \Rightarrow a = - \frac{g}{R} y$

This acceleration will be towards the centre of the earth.

So the component of acceleration towards the center of the tunnel.

$\begin{aligned} a^{'} = a \sin θ = (- \frac{g}{R} y) (\frac{x}{y}) = - \frac{g}{R} x \\ Comparing with a^{'} = - ω^{2} x \\ ω^{2} = \frac{q}{R} \Rightarrow ω = \sqrt{\frac{q}{R}} \Rightarrow T = 2 π \sqrt{\frac{R}{g}} = 84.6 \min \end{aligned}$

Note: The time period of oscillation is the same in both cases whether the tunnel is along a diameter or along the chord.

Solved Examples Based on Motion of a Ball in Tunnel Through the Earth

Example 1: The mass and diameter of a planet are twice those of Earth. The period of oscillation of the pendulum on this planet will be (If its second's pendulum on earth)

1) $\frac{1}{\sqrt{2}} s$
2) $2 \sqrt{2} s$
3) 2 s
4) $\frac{1}{2} s$

Solution:

The Time period of a ball through a tunnel in the earth

$\begin{aligned} T = 2 π \sqrt{\frac{R}{g}} \\ = & 84.6 \min \end{aligned}$
wherein
$R =$ Radius of earth
$g =$ acceleration due to gravity
As we know $g = \frac{G M}{R^{2}}$
Given $M_{p} = 2 M_{e}$
and $D_{p} = 2 D_{e} \Rightarrow R_{p} = R_{e}$
$\Rightarrow \frac{g_{earth}}{g_{planet}} = \frac{M_{e}}{M_{p}} \times \frac{R_{p}^{2}}{R_{e}^{2}} \Rightarrow \frac{g_{earth}}{g_{planet}} = 2$

Also,
$\begin{aligned} T α \frac{1}{\sqrt{g}} \Rightarrow \frac{T_{e}}{T_{p}} = \sqrt{\frac{g_{p}}{g_{e}}} \Rightarrow \frac{2}{T_{p}} = \sqrt{\frac{1}{2}} \\ \Rightarrow & T_{p} = 2 \sqrt{2} s \end{aligned}$

Hence, the answer is the option (2).

Example 2: If a body is released into a tunnel dug across the diameter of the earth, it executes simple harmonic motion with time period

1) $T = 2 π \sqrt{\frac{R_{e}}{g}}$
2) $T = 2 π \sqrt{\frac{2 R_{e}}{g}}$
3) $T = 2 π \sqrt{\frac{R_{e}}{2 g}}$
4) $T = 2 s$

Solution:

The Time period of a ball through the tunnel in the earth

$T = 2 π \sqrt{\frac{R}{g}}$
wherein
$R =$ The radius of the earth's
$g =$ acceleration due to gravity

Hence, the answer is the option (1).

Example 3: A ball is released from the surface of the earth along a tunnel such that the tunnel is along the diameter of the earth, then the motion of this ball will have time period equal to T₁ and if another ball of twice the mass of the ball in the first case is released along any of the chord of the earth and its motion is having a time period of T_2.Then which of the given relations will be true?

1) $T_{2} = \sqrt{2} \cdot T_{1}$ because the mass of the second case is two times of the first case
2) $T_{2} = T_{1}$
3) Can not say as the distance of the chord is not given from the centre of the earth.
4) $\sqrt{2} \cdot T_{1} = T_{2}$

Solution:

Case-1. When the tunnel is along a diameter and the ball is released from the surface, then the time period of motion,

$T_{1} = 2 π \sqrt{\frac{R}{g}}$

Case 2. When the tunnel is along a chord and the ball is released from the surface, then the time period of motion

$T_{2} = 2 π \sqrt{\frac{R}{g}}$

The time period of oscillation is the same in both cases whether the tunnel is along the diameter or along the chord. It does not depend on the mass. So, $T_{2} = T_{1}$ .

Hence, the answer is the option (2).

Example 4: Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is 100 g. The time period of the motion of the particle will be (approximately) ( Take $g = 10 m s^{- 2}$ , the radius of earth $= 6400 km$ )

1) 12 hours

2) 1 hour 40 minutes

3) 24 hours

4) 1 hour 24 minutes

Solution:

Inside Earth, force is given by $F = - \frac{G M_{e} m x}{R_{e}^{3}}$
And $g_{0} ($ on the surface of the earth $) = \frac{G M_{e}}{R_{e}^{2}}$
$\begin{aligned} ∴ F = - \frac{g_{0} m}{R_{e}} x \\ \Rightarrow a = - \frac{g_{0}}{R_{e}} x \\ ω = \sqrt{\frac{g_{0}}{R_{e}}} \\ \Rightarrow T = 2 π \sqrt{\frac{R_{e}}{g_{0}}} = 2 π \sqrt{\frac{6400 \times 10^{3}}{10}} = 2 \times 3.13 \times 8 \times 10^{2} \sec = 5024 \sec = 1.4 hr \\ T = 1.4 hr = 1 hr 24 minutes \end{aligned}$

Hence, the answer is the option (4).

Summary

The motion of a ball in a tunnel through the Earth can be understood as a form of simple harmonic motion, driven by gravitational forces. Whether the tunnel is along the Earth's diameter or along a chord, the ball oscillates back and forth within the same time period, which is independent of the ball's mass. This concept highlights the uniformity of gravitational effects and the principles of harmonic motion, demonstrating how they apply even in such extreme theoretical scenarios.

Frequently Asked Questions (FAQs)

1. What would happen if you dropped a ball through a tunnel that goes straight through the center of the Earth?

If you dropped a ball through a tunnel going straight through Earth's center, it would oscillate back and forth between the entry points on opposite sides of the planet. The ball would accelerate towards the center due to gravity, reach maximum speed at the center, then slow down as it climbs up the other side. This motion would repeat, creating a simple harmonic oscillation, assuming no air resistance or other friction.

2. Why doesn't the ball just stay at the Earth's center when it reaches that point?

The ball doesn't stay at Earth's center because of its inertia. As it falls, it gains kinetic energy and reaches maximum speed at the center. This momentum carries it past the center point. The ball then starts to slow down as gravity pulls it back, but it has enough energy to reach the opposite side of the Earth before falling back again.

3. Would the motion of the ball be considered simple harmonic motion?

Yes, the motion of the ball through Earth would be considered simple harmonic motion. The gravitational force acting on the ball is always directed towards the center of the Earth and is proportional to the ball's displacement from the center. This results in a restoring force that satisfies the conditions for simple harmonic motion.

4. How does air resistance affect the ball's motion in this thought experiment?

In reality, air resistance would significantly affect the ball's motion. It would cause the ball to lose energy with each oscillation, gradually reducing the amplitude of its motion. Eventually, the ball would come to rest at Earth's center. However, in the ideal thought experiment, we typically ignore air resistance to focus on the gravitational effects.

5. How does the ball's speed vary as it moves through the Earth?

The ball's speed varies continuously as it moves through Earth. It starts from zero at the surface, increases as it falls towards the center, reaches its maximum speed at Earth's center, then decreases as it moves towards the opposite surface. This pattern repeats for each oscillation, with the speed always being greatest at the center and zero at the extremes.

6. How long would it take for the ball to complete one full oscillation through the Earth?

The time for one complete oscillation through Earth, known as the period, would be approximately 84 minutes. This is calculated using the formula T = 2π√(R/g), where R is Earth's radius and g is the acceleration due to gravity at Earth's surface. Interestingly, this period is independent of the ball's mass or the tunnel's specific path, as long as it passes through Earth's center.

7. What would the velocity-time graph look like for the ball's motion through Earth?

The velocity-time graph for the ball's motion would be a sine wave. The velocity would be zero at the entry points (where the ball changes direction), reach its maximum (positive or negative) at the center of the Earth, and then decrease back to zero at the opposite side. This pattern would repeat for each oscillation.

8. How would the ball's motion differ if Earth had a uniform density throughout?

If Earth had a uniform density throughout, the ball's motion would still be simple harmonic, but the period of oscillation would be different. In a uniform density Earth, the gravitational force would increase linearly from zero at the center to a maximum at the surface, rather than being nearly constant near the surface as it is in the real Earth. This would result in a longer period of oscillation.

9. What would happen if the tunnel were filled with a fluid instead of being empty?

If the tunnel were filled with a fluid, the ball's motion would be significantly altered. The fluid would exert a drag force on the ball, causing it to lose energy with each oscillation. The motion would no longer be simple harmonic. Instead, the ball would undergo damped oscillations, with the amplitude decreasing over time until the ball eventually comes to rest at Earth's center.

10. What would happen if you tried to stop the ball at Earth's center?

Stopping the ball at Earth's center would require applying a force equal and opposite to the ball's momentum. Once stopped, the ball would remain at the center, as there would be no net gravitational force acting on it there. However, any slight displacement from the center would cause the ball to start oscillating again, as Earth's center is an unstable equilibrium point.

11. What would happen to the period of oscillation if the tunnel didn't pass through Earth's center?

If the tunnel didn't pass through Earth's center, the period of oscillation would change. For any chord through a uniform sphere, the period of oscillation is the same as if the ball were oscillating through the center. However, the period would be shorter than the 84 minutes for a diameter tunnel, as the effective length of the oscillation path is reduced.

12. What would happen if you released the ball with an initial velocity instead of just dropping it?

If you released the ball with an initial velocity, it would affect the amplitude of the oscillation but not the period. A horizontal initial velocity would cause the ball to follow an elliptical path instead of a straight line. A downward initial velocity would increase the amplitude, while an upward initial velocity would decrease it. However, the 84-minute period would remain unchanged.

13. How would the ball's motion be affected if Earth were not in isolation, but orbiting the Sun?

Earth's orbit around the Sun would not significantly affect the ball's motion through Earth. The entire Earth-ball system is in free fall around the Sun, so the ball's motion relative to Earth remains unchanged. This is an application of Einstein's equivalence principle, which states that the effects of gravity are indistinguishable from the effects of acceleration.

14. What would happen if you tried to communicate with the ball using radio waves as it passed through Earth's center?

Communicating with the ball using radio waves as it passes through Earth's center would be challenging. The dense material of Earth's core would severely attenuate or block most radio signals. Additionally, the rapid motion of the ball through regions of varying density could cause significant Doppler shifts in any signals that do get through.

15. What would happen if you tried to measure the ball's position using GPS as it moved through Earth?

Using GPS to measure the ball's position inside Earth would be impossible. GPS signals cannot penetrate deep into the Earth. They rely on line-of-sight transmission from satellites, so they would only work near the surface. Alternative methods, such as accelerometers or gyroscopes, would be needed to track the ball's motion inside Earth.

16. What would happen if the ball were replaced with a quantum particle?

If the ball were replaced with a quantum particle, its behavior would be described by quantum mechanics rather than classical mechanics. The particle would have a wave function extending throughout the tunnel, with the highest probability of being found at the classical turning points (Earth's surface) and the lowest at the center. It would exhibit quantum tunneling, potentially being found outside the classical boundaries of oscillation.

17. What would happen if the ball were spinning as it fell through the tunnel?

If the ball were spinning as it fell through the tunnel, it would continue to spin due to conservation of angular momentum. The spin itself wouldn't significantly affect the ball's translational motion through Earth. However, if the tunnel had any air or if the ball were to touch the sides, the spin could induce additional forces (like the Magnus effect) that might slightly alter the ball's path.

18. What would happen if the ball were replaced with a fluid?

If the ball were replaced with a fluid, the behavior would be quite different. Instead of oscillating as a single unit, the fluid would spread out along the tunnel. It would still experience oscillatory motion, with the center of mass of the fluid following a similar pattern to a solid ball. However, the fluid would also experience internal motions, waves, and potentially turbulence, making the overall behavior much more complex.

19. How would the ball's motion be affected by tidal heating from Earth's gravitational field?

Tidal heating from Earth's gravitational field would have a negligible effect on the ball's motion. Tidal heating is more significant for large bodies or bodies in elliptical orbits. For a small ball moving through Earth, the effect would be too small to noticeably influence its motion. The ball might heat up slightly due to this effect, but not enough to alter its trajectory.

20. How does the gravitational force on the ball change as it moves through the Earth?

The gravitational force on the ball varies linearly with its distance from Earth's center. It's maximum at the surface, decreases linearly to zero at the center, then increases linearly (in the opposite direction) as the ball moves towards the other side. This linear relationship between force and displacement is what makes the motion simple harmonic.

21. Would the ball's motion be affected if Earth wasn't a perfect sphere?

Yes, if Earth wasn't a perfect sphere, the ball's motion would be affected. Earth's actual shape (an oblate spheroid) and variations in density would cause slight deviations from perfect simple harmonic motion. The period of oscillation might vary slightly depending on the tunnel's orientation relative to Earth's axis.

22. How does the concept of potential energy apply to the ball's motion through Earth?

The concept of potential energy is crucial in understanding the ball's motion. At the surface, the ball has maximum gravitational potential energy. As it falls, this potential energy converts to kinetic energy. At the center, all potential energy has been converted to kinetic energy. As the ball rises on the other side, kinetic energy converts back to potential energy. This energy conversion cycle repeats with each oscillation.

23. How does the acceleration of the ball change during its journey through Earth?

The acceleration of the ball changes continuously during its journey. It's maximum at the surface, decreases linearly to zero at the center, then increases linearly in the opposite direction as the ball moves towards the other side. This changing acceleration is what causes the ball to slow down as it approaches the surface and speed up as it falls back towards the center.

24. Would the ball's motion be affected by Earth's rotation?

Earth's rotation would have a small effect on the ball's motion. The Coriolis effect would cause the ball to deviate slightly from a straight path, making it trace out an ellipse instead. However, for the purposes of most thought experiments, this effect is usually considered negligible compared to the dominant gravitational forces.

25. What would happen to the ball's motion if Earth suddenly disappeared, leaving only its gravitational field?

If Earth suddenly disappeared but its gravitational field remained unchanged, the ball would continue its oscillatory motion as before. The gravitational field, not the physical presence of Earth, is what determines the ball's motion. However, this scenario is purely hypothetical as the gravitational field is a property of the mass distribution that creates it.

26. How does the kinetic energy of the ball vary during its motion through Earth?

The kinetic energy of the ball varies continuously during its motion. It's zero at the surface (entry points), increases as the ball falls towards the center, reaches its maximum at Earth's center, then decreases as the ball moves towards the opposite surface. This pattern repeats for each oscillation, with kinetic energy always being greatest at the center and zero at the extremes.

27. Would the ball's motion be affected by tidal forces from the Moon or Sun?

Tidal forces from the Moon and Sun would have a negligible effect on the ball's motion through Earth. These forces are much weaker than Earth's gravity and would cause only tiny perturbations in the ball's path. For practical purposes in this thought experiment, they can be ignored.

28. How would the ball's motion be affected by Earth's magnetic field?

Earth's magnetic field would have a negligible effect on the ball's motion unless the ball were magnetic or electrically charged. For a typical non-magnetic, non-conductive ball, the magnetic field's influence would be far too weak compared to the gravitational forces to noticeably alter the ball's path.

29. How would the ball's motion be affected by relativistic effects?

For a ball moving through Earth, relativistic effects would be negligible. The maximum speed the ball would reach (about 8 km/s at Earth's center) is far too low for relativistic effects to be significant. These effects only become noticeable at speeds approaching that of light, which is about 300,000 km/s.

30. How would the ball's motion be affected if Earth were expanding or contracting?

If Earth were expanding or contracting, the ball's motion would be affected. An expanding Earth would cause the period of oscillation to increase over time, while a contracting Earth would cause it to decrease. The amplitude of oscillation would also change. However, any realistic rate of Earth's expansion or contraction would be too slow to noticeably affect the ball's motion over a few oscillations.

31. How would the ball's motion be affected by earthquakes or other geological activities?

Earthquakes or other geological activities could potentially disrupt the ball's motion. They might cause vibrations in the tunnel walls, potentially imparting additional energy to the ball or altering its path. However, for most theoretical discussions of this thought experiment, such geological effects are typically ignored to focus on the idealized gravitational behavior.

32. What would happen if multiple balls were dropped into the tunnel at different times?

If multiple balls were dropped into the tunnel at different times, they would all oscillate with the same period but different phases. They would pass through the center at different times and could potentially collide with each other if their paths intersected. The motion of each ball would be independent of the others, assuming they don't collide.

33. How would the ball's motion be affected by changes in Earth's mass distribution over time?

Changes in Earth's mass distribution over time, such as those caused by plate tectonics or core dynamics, would affect the ball's motion. However, these changes occur over geological timescales (millions of years) and would be imperceptible over the course of a few oscillations. For all practical purposes in this thought experiment, Earth's mass distribution can be considered constant.

34. How would the ball's motion be affected by the curvature of spacetime near Earth?

The curvature of spacetime near Earth, as described by general relativity, does affect the ball's motion, but the effect is extremely small for this scenario. It causes a very slight deviation from perfect simple harmonic motion, but this deviation is so minute that it can be ignored for all practical purposes in this thought experiment.

35. What would happen if the tunnel were not perfectly straight?

If the tunnel were not perfectly straight, the ball's motion would be more complex. It would no longer follow simple harmonic motion. The ball would likely follow a more complicated path, potentially bouncing off the tunnel walls. The period of oscillation would likely increase, and the motion might become chaotic depending on the tunnel's shape.

36. How would the ball's motion be affected by Earth's oblateness?

Earth's oblateness (its slight flattening at the poles) would have a small effect on the ball's motion. For a pole-to-pole tunnel, the period would be slightly shorter than for an equatorial tunnel, due to the difference in Earth's radius. The motion would also deviate slightly from perfect simple harmonic motion, but these effects would be minor for most practical considerations.

37. What would happen if the ball were made of a material that could withstand the extreme conditions inside Earth?

If the ball were made of a material that could withstand the extreme conditions inside Earth, its motion would remain largely unchanged. The ball's composition doesn't affect its gravitational interaction with Earth. However, such a ball might provide valuable data about Earth's interior conditions if it could be retrieved or if it could transmit information during its journey.

38. How would the ball's motion be affected by the presence of large underground cavities or variations in rock density?

Large underground cavities or variations in rock density would cause small perturbations in the ball's motion. These inhomogeneities would create local variations in the gravitational field, causing the ball to deviate slightly from its ideal path. However, for a planet-wide tunnel, these effects would generally average out and wouldn't significantly alter the overall oscillatory nature of the motion.

39. How would the ball's motion be affected by Earth's thermal gradient?

Earth's thermal gradient wouldn't directly affect the ball's motion, as gravity is the dominant force. However, if the ball were not perfectly insulated, it would heat up as it approached Earth's hot core and cool down as it moved back towards the surface. This heating and cooling could cause thermal expansion and contraction of the ball, potentially affecting its size and, to a very small degree, its motion.

40. What would happen if the tunnel were evacuated of all air?

If the tunnel were evacuated of all air, the ball's motion would more closely approximate the ideal simple harmonic motion described in the thought experiment. Without air resistance, the ball would continue to oscillate indefinitely (ignoring other forms of friction). This scenario is often assumed in theoretical discussions to simplify the analysis and focus on gravitational effects.

41. How would the ball's motion be affected by the Coriolis effect in a rotating reference frame?

In a rotating reference frame (like Earth's surface), the Coriolis effect would cause the ball to deviate from a straight path. For a north-south tunnel, the ball would appear to curve slightly to the west when moving southward and to the east when moving northward. For an east-west tunnel, the effect would cause the ball to appear to rise and fall slightly. However, these effects are small compared to the primary gravitational motion.