Question 39.1: What Is the Period of the Pendulum? The period of a pendulum......
What Is the Period of the Pendulum?
The period of a pendulum is measured to be 3.00 \mathrm{~s} in the reference frame of the pendulum. What is the period when measured by an observer moving at a speed of 0.960 \mathrm{c} relative to the pendulum?
Conceptualize Let’s change frames of reference. Instead of the observer moving at 0.960 c, we can take the equivalent point of view that the observer is at rest and the pendulum is moving at 0.960 c past the stationary observer. Hence, the pendulum is an example of a clock moving at high speed with respect to an observer.
Categorize Based on the Conceptualize step, we can categorize this problem as one involving time dilation.
Analyze The proper time interval, measured in the rest frame of the pendulum, is \Delta t_{p}=3.00 \mathrm{~s}.
Use Equation 39.7 to find the dilated time interval:
\begin{aligned}\Delta t & =\gamma \Delta t_{p}=\frac{1}{\sqrt{1-\frac{(0.960 c)^{2}}{c^{2}}}} \Delta t_{p}=\frac{1}{\sqrt{1-0.9216}} \Delta t_{p} \\& =3.57(3.00 \mathrm{~s})=10.7 \mathrm{~s}\end{aligned}
\Delta t={\frac{\Delta t_{\rho}}{\sqrt{1-{\frac{v^{2}}{c^{2}}}}}}=\gamma\,\Delta t_{\rho} (39.7)
Finalize This result shows that a moving pendulum is indeed measured to take longer to complete a period than a pendulum at rest does. The period increases by a factor of \gamma=3.57.