Understanding the Concepts
* Time Dilation: Special relativity tells us that time passes slower for an object moving at a significant fraction of the speed of light (c) relative to a stationary observer. This effect is known as time dilation.
* Pendulum Period: The period of a pendulum is the time it takes for one complete swing.
Derivation
1. Time Dilation Formula: The time dilation formula from special relativity is:
```
t' = t / sqrt(1 - v^2/c^2)
```
Where:
* t' is the time measured by the moving observer
* t is the time measured by the stationary observer
* v is the relative velocity between the observer and the pendulum
* c is the speed of light
2. Applying to Pendulum Period: The period of the pendulum is the time it takes for one complete swing. Let:
* T be the period of the pendulum as measured by a stationary observer
* T' be the period of the pendulum as measured by the observer moving at 0.95c
Then, using the time dilation formula:
```
T' = T / sqrt(1 - (0.95c)^2/c^2)
```
3. Simplifying the Equation:
```
T' = T / sqrt(1 - 0.9025)
```
```
T' = T / sqrt(0.0975)
```
```
T' ≈ T / 0.312
```
Conclusion
The period of the pendulum as measured by the observer moving at 0.95c will be approximately 3.2 times longer than the period measured by a stationary observer.
Important Note: This calculation assumes the pendulum is at rest in the stationary frame of reference. If the pendulum is also moving with respect to the stationary observer, the calculation would be more complex.
