Ch. 36 - The Special Theory of Relativity

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 36 - The Special Theory of Relativity

Problem 66

Chapter 35, Problem 66

An atomic clock is taken to the North Pole, while another stays at the Equator. How far will they be out of synchronization after 1.5 years has elapsed? [Hint: Use the binomial expansion, Appendix A–2.]

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Identify the effect of Earth's rotation on time dilation. The clock at the Equator moves faster relative to the clock at the North Pole due to Earth's rotation, which affects the time measured by each clock due to relativistic effects.

Use the formula for time dilation under special relativity, which is \$\Delta t' = \Delta t \sqrt{1 - \frac{v^2}{c^2}}\$, where \$\Delta t\$ is the proper time interval, \$\Delta t'\$ is the dilated time interval, $\v$\\ is the velocity, and $\c$\\ is the speed of light.

Calculate the velocity ($\v$\\) of the clock at the Equator due to Earth's rotation. The velocity can be found using the radius of the Earth at the Equator and the rotational period of the Earth (24 hours).

Apply the binomial expansion to the time dilation formula for small values of \$\frac{v^2}{c^2}\$ to simplify the calculation. The binomial expansion of \$\sqrt{1 - x}\$ for small $\x$\\ is approximately \\1 - $\frac{x}{2}$\\.

Calculate the difference in time (\$\Delta t' - \Delta t\$) between the two clocks after 1.5 years, using the simplified time dilation formula and the velocity calculated in step 3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Time Dilation

Time dilation is a phenomenon predicted by Einstein's theory of relativity, where time passes at different rates for observers in different gravitational fields or relative velocities. In this scenario, the atomic clock at the North Pole experiences a stronger gravitational pull compared to the one at the Equator, leading to a difference in the passage of time between the two clocks.

Gravitational Potential

Gravitational potential refers to the potential energy per unit mass at a point in a gravitational field. The North Pole is closer to the center of the Earth than the Equator due to the Earth's oblate shape, resulting in a higher gravitational potential at the Equator. This difference affects the rate at which time passes for the clocks located at these two points.

Binomial Expansion

The binomial expansion is a mathematical technique used to approximate expressions involving powers of binomials. In the context of this problem, it can be used to simplify calculations related to the small differences in time experienced by the two clocks due to their different gravitational potentials, allowing for an easier estimation of their synchronization discrepancy over 1.5 years.

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