Question

A chin plate has a round hole whose diameter in its rest frame is \(D\). The plate is parallel to the ground and moving upward, in the \(+y\) direction, relative to the ground. A thin round disk whose diameter in its rest frame is \(D\) is also parallel to the ground but moving in the \(+x\) direction relative to it. In the frame of the ground, the plate and disk are on course so that the centers of the hole and disk will at some point coincide. The disk is contracted, but the hole in the plate is not, so the disk will pass through the hole. Now consider the frame of the disk. The disk is of diameter \(D\). but the hole is contracted. Can the disk pass through the hole, and if so, how?

Short answer

Yes, the disk can pass through the hole both from the disk's and also from the plate's frame of reference. The hole doesn't contract in either of those frames because it's moving perpendicularly to its diameter, and Lorentz-FitzGerald contraction doesn't occur in that case. This problem is often cited as a seeming paradox in special relativity because it's easy to mistakenly believe that Lorentz contraction depends on relative motion rather than observer.

Step-by-step solution

Analyzing the situation in the plate's frame of reference

The plate is stationery in its frame of reference. The disk moving in horizontally in \(+x\) direction. Since the motion is perpendicular to the diameter of the disk, Lorentz-FitzGerald contraction does not occur. Thus the disk remains of diameter \(D\) in the plate's frame. The plate itself being stationary, its hole is also of diameter \(D\). Hence, the disk can pass through the hole.

Analyzing the situation in the disk's frame of reference

In the disk's frame of reference, the disk is stationery and therefore retains its original diameter \(D\). The plate however, is moving upward in \(+y\) direction, perpendicular to the diameter of the hole. Therefore, the diameter of the hole is not contracted, it remains \(D\). Thus, even in the disk's frame, the disk can pass through the hole.

Understanding the seeming paradox

This problem seems to create a paradox. You might expect that since Lorentz-FitzGerald contraction only affects dimensions parallel to the direction of motion, then in the frame of the moving disk, the hole, which is moving in a perpendicular direction, would contract. However, Lorentz-FitzGerald contraction is relative to the observer, not to other objects. In the frame of the ground, both the disk and its hole contract, so the disk can pass through the hole. In the frame of either the disk or the hole, they are stationary and do not contract, so again, the disk can pass through the hole. Therefore, there is no paradox.

Key concepts

Relativity

Relativity is an essential theory in physics developed by Albert Einstein. It fundamentally changed our understanding of time and space. Understanding relativity helps us solve many intriguing problems in physics, such as the one involving the disk and plate moving relative to each other.

Relativity states that the laws of physics are the same for all observers, regardless of their motion. This means that the measurements of time and space are relative, not absolute. Two observers moving in different frames may perceive an event differently, such as the passage of a disk through a hole.

In our problem, both the disk and the plate perceive themselves as having a diameter of \(D\) in their own reference frames. The fascinating part comes into play when we consider how these objects appear to contract when viewed from different frames, an idea explained by the Lorentz-FitzGerald contraction. Relativity helps us understand and resolve these counterintuitive phenomena, demonstrating the beautiful complexity of our universe.

Reference Frame

A reference frame is essentially a point of view from which we measure physical events. In physics, it is crucial because it defines what is happening to objects in motion relative to each other. Let's unpack this using our disk and plate scenario.

In the frame of reference of the plate, the plate is stationary, and the disk moves horizontally. In this frame, since the motion of the disk is perpendicular to its diameter, there is no Lorentz-FitzGerald contraction, and the disk maintains its diameter of \(D\). Consequently, the disk easily passes through the hole.

However, if we shift our perspective to the frame of reference of the disk, the disk is stationary, and now the plate moves upward. Again, since motion is perpendicular to the diameter of the hole, it remains uncontracted at a diameter \(D\). This shows how the concept of reference frames allows one to understand the dynamics of movement and contractions without invoking any paradox.

Paradox Resolution

When dealing with relativity, apparent paradoxes can arise, leading to confusion. However, these paradoxes often resolve upon closely examining the physics principles involved. This exercise involving the disk and plate presents such an apparent paradox.

Initially, one might think there is a paradox. Since movement impacts how objects contract based on their reference frames, the question arises: "Why doesn't the hole contract in the disk's frame?" The clarification comes from understanding the directional aspect of Lorentz-FitzGerald contraction, which only affects the dimension parallel to the direction of motion.

In this problem, the motion is perpendicular to the diameter of both the hole and the disk in their respective frames. Thus, neither contracts, allowing the disk to pass seamlessly through the hole in both frames. This analysis shows how apparent contradictions disappear with a proper understanding of relativity and reference frames, paving the way for clear logical conclusions.