Question

A meterstick is glued to the wall with its \(100 \mathrm{~cm}\) end farther to the right, in the positive direction. It has a clock at its center and one on each end. You walk by A meterstick is glued to the wall with its 100 em end farther to the right, in the positive direction. It has a clock at its center and one on each end. You walk by the meterstick in the positive direction at speed \(v\). (a) When you reach the center clock, it reads \(0 .\) What do the other two read at this instant in your frame. (b) You instantly reverse direction. The clock at the center is still reading 0 and so is yours. What do the others read? (c) How does this relate to the twin paradox?

Short answer

In both cases, the end clocks read zero in your frame due to the effects of time dilation. The connection with the Twin Paradox lies in the concept of time dilation; the end clocks can be viewed as the traveling twin who ages less than the stationary twin.

Step-by-step solution

Time Dilation for Forward Direction

Firstly, let's analyze the situation when moving in a forward direction. Here the time dilation concept comes into action. It refers to the idea that time moves slower when observed by someone who is moving compared to an observer who is stationary. In our case, the student walking past the meterstick will witness the clocks ticking slower than the clocks at rest with the meterstick. Therefore, when you reach the center clock and see it read zero, the other two clocks, from your perspective, will also read zero because you, the meterstick and the clocks were all initially in the same frame.

Time Dilation for Reverse Direction

Now consider the situation when direction is instantly reversed. Even in this case, the time dilation concept is still relevant. Similar to the previous case, due to time dilation, the clocks on the ends would have 'aged' less than the center clock in your frame. Hence, they will continue to read zero.

Link with the Twin Paradox

The Twin Paradox is a thought experiment in special relativity, where one twin undertakes a journey into space in a high-speed rocket and comes back home to find that the other twin has aged more. This problem is related to the Twin Paradox as both situations involve the concepts of time dilation. In this scenario, the clocks can be thought of as the twins. The clocks on the ends of the meterstick could be viewed as the traveling twin (since from your frame they were in motion while you were temporarily at rest when you reversed directions), who ages less compared with the twin who stayed at home (i.e., you, or equivalently the clock at the center).

Key concepts

Time Dilation

Imagine wearing a wristwatch that ticks away time. Now, if you were to run at a significant fraction of the speed of light past a row of stationary clocks, you'd surprisingly find that your watch records less time than the clocks you speed by. This mind-bending phenomenon is called time dilation, a core concept within the realm of Albert Einstein's special theory of relativity.

Time dilation expresses that time, just like physical space, isn't absolute but relative. It essentially means that clocks moving at high speeds in relation to each other will not agree on the measurement of time; the faster a clock moves, the slower it appears to tick if you're observing it from a different velocity. This is not a product of the clock's mechanics but a fundamental aspect of the fabric of reality itself.

In exercises where students might see questions about moving clocks or observing passing events while in motion, they are encountering practical implications of time dilation. For instance, as you walk past the centrally-placed clock, it can lead to a disconcertingly non-intuitive result: all of the clocks appear synchronized to you initially because they were all at rest relative to each other before you began moving.

Twin Paradox

The Twin Paradox is more than just a plot for a science fiction narrative; it's a real consequence of time dilation in the special theory of relativity. Fundamentally, it's a thought experiment involving identical twins where one twin travels on a spaceship at a speed close to the speed of light, while the other twin remains on Earth. Upon the spacefaring twin’s return, the Earth-bound twin has aged more—a paradox, if you will, since from each twin's perspective, the other should be the one moving.

This paradox highlights that time dilation isn't symmetrical and teases out implications of simultaneity and relative motion. Relating such an imaginative scenario to schoolwork on the topic, like reversing the direction past a meterstick, helps to make the paradox more palpable to students. By attributing the role of the traveling twin to the clocks at the ends, it becomes clearer why, from the walker's perspective, these 'twins' haven't aged, though the stationary observer has remained in the same spot, untouched by the journey.

Special Theory of Relativity

The special theory of relativity overturned centuries of assumptions about the absolutes of time and space. It postulates that the laws of physics are the same for all non-accelerating observers, and that the speed of light in a vacuum is constant, regardless of the motion of the light source or observer. What this translates to, in everyday language, is that there is no single universal 'now' or fixed space: they're relative to the observer's motion. What one perceives as simultaneous or lengths of lengths and times can differ for another observer moving at a different speed.

School exercises often ask students to apply this theory to understand phenomena like time dilation or the Twin Paradox. When a problem involves walking past a meterstick with clocks, it encapsulates much of what special relativity is about: from different frames of reference (your frame as you walk vs. the meterstick’s rest frame), the reported times on the clocks will be different due to relative motion. The beauty of special relativity lies within such counterintuitive outcomes, challenging our innate perceptions and broadening understanding of the universe.