Question

Observer A sees a pendulum oscillating back and forth in a relativistic rocket and measures its period to be T_A. Observer B moves along with the rocket and measures the period of the pendulum to be T_B. What is true about these two time measurements?

	TA < TB
	TA > TB
	TA = TB
	TA could be greater or smaller than TB depending on the direction of the motion.
	None of the above is correct.

Answer 1

Concepts and reason

The concept used to solve this problem is time dilation.

Initially, find the time period measured by the observer A for the pendulum oscillating in which the rocket is moving using the concept of relativistic equation for time period.

Then, find the time period of the pendulum measured by the observer B who moves along with the rocket.

Finally, use the concept of time dilation for the observer A and observer B to find the true measurement is taken by the observer A or B by relating the time period of the pendulum.

Fundamentals

According to the theory of relativity, time dilation is a difference in the elapsed time measured by two observers either due to a velocity difference relative to each other. This can be expressed as follows:

Here, is the time period of the moving observer of the clock, is the time period of the observer moving by itself relative to the clock, is the velocity of the object, and is the velocity of light.

The expression for the time dilation relating the observers A and B is as follows:

Here, is the time period measured by the observer A in which the rocket is moving, is the speed of the rocket, is the speed of the light, and is the time period measured by observer B stationary with the observer A.

The incorrect options are as follows:

•

•

• could be greater or smaller than depending on the direction of the motion.

• None of the above options are correct.

The time measurement of an observer at rest with respect to the time measurement moving observer is more. So, the time measurement of the rest observer is called time dilation.

The time period of the pendulum measured by the observer B in the rocket is called proper time.

The time measurement of an observer A is more than the time measurement of the observer B because observer A is at rest; it is shown in the below equation.

The time dilation relating the observers A and B is as follows:

…… (1)

The velocity of the object is less than the velocity of light ().

The above Equation (1) becomes

Hence, the correct option is as follows:

•

Ans:

Thus, the true equation for the two time measurements from the observers A and B is .