Question

A bead is tied on one end of a stiff rope of length 1 m. With the other end of the rope as the center, the rope is rotated in such a way that the bead completes 10 revolutions per second. The centripetal acceleration of the bead is

• A. $400\pi^2$ m/s$^2$
• B. $200\pi^2$ m/s$^2$
• C. 400 m/s$^2$
• D. 200 m/s$^2$
• E. 100 m/s$^2$

Hint:

Always keep track of whether your answer needs to be in terms of $\pi$ or a numerical approximation before finalizing your choice.

Answer 1

The Correct Option is C

Concept:
Centripetal acceleration ($a_c$) for an object in uniform circular motion can be calculated using angular velocity ($\omega$) and radius ($r$): \[ a_c = \omega^2 r \]

Step 1: Convert frequency to angular velocity.
Given frequency ($f$) = 10 revolutions/second. \[ \omega = 2\pi f = 2\pi \times 10 = 20\pi \text{ rad/s} \]

Step 2: Calculate acceleration with $\pi$.
Using $r = 1$ m: \[ a_c = (20\pi)^2 \times 1 = 400\pi^2 \text{ m/s}^2 \]

Step 3: Approximation for standard numerical answers.
In many physics problems, the approximation $\pi^2 \approx 10$ is used to simplify results: \[ a_c \approx 400 \times 10 = 4000 \text{ m/s}^2 \]