What is the speed of a body vibrating at 50 cyclic per second

Analysis

The question asks for the "speed" of a vibrating body given its frequency. In the context of simple harmonic motion (SHM) and waves, the term "speed" here refers to the angular velocity (or angular frequency), which is measured in radians per second ($\text{rad s}^{-1}$).

Concept

For a body vibrating with a frequency ($f$), the angular velocity ($\omega$) is given by the formula:

Where:

• $\omega$ = angular velocity (in $\text{rad s}^{-1}$)
• $f$ = frequency (in cycles per second or Hertz)
• $\pi$ = a constant (approximately 3.142)

Solution

1. Identify the given value: The frequency ($f$) is given as 50 cycles per second (or 50 Hz).

2. Apply the formula: $$\omega = 2 \times \pi \times f$$ $$\omega = 2 \times \pi \times 50$$

3. Calculate the result: $$\omega = 100\pi \text{ rad s}^{-1}$$

Conclusion

Comparing our result to the provided options, the correct value is $100\pi \text{ rad s}^{-1}$. This matches option A.

Exam Tip

Always pay attention to the units in the options. If the question asks for "speed" in a vibration context and the options have $\text{rad s}^{-1}$, it is a clear signal that you are being asked to calculate the angular frequency ($\omega$), not the linear velocity ($v = \omega r$). Don't let the terminology confuse you!