Question

A body performing uniform circular motion of radius $R$ has frequency $n$. Its centripetal acceleration is

• A. $8\pi^2 nR^2$
• B. $4\pi^2 n^2 R$
• C. $4\pi^2 n^2 R^2$
• D. $8\pi^2 n^2 R$

Hint:

You can use standard dimensional analysis to double-check your answer quickly! Acceleration must have dimensions of $\text{LT}^{-2}$. Frequency $n$ has units of $\text{s}^{-1}$ ($\text{T}^{-1}$) and radius $R$ has units of $\text{m}$ ($\text{L}$). Checking option (B): $n^2 R \rightarrow (\text{s}^{-2})(\text{m}) = \text{m/s}^2$. This matches the dimensions of acceleration perfectly!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the algebraic expression for the centripetal acceleration of an object undergoing Uniform Circular Motion (U.C.M.) along a trajectory of radius $R$ at a rotational frequency $n$.

Step 2: Key Formula or Approach:
1. The standard formula for centripetal acceleration ($a_c$) in terms of linear velocity ($v$) or angular velocity ($\omega$) is: $$a_c = \frac{v^2}{R} = \omega^2 R$$ 2. Relate the angular velocity $\omega$ to the rotational frequency $n$: $$\omega = 2\pi n$$

Step 3: Detailed Explanation:
Let's substitute the frequency relation $\omega = 2\pi n$ directly into the angular form of the centripetal acceleration formula: $$a_c = \omega^2 R$$ $$a_c = (2\pi n)^2 R$$ Expanding the squared term gives: $$a_c = 4\pi^2 n^2 R$$

Step 4: Final Answer:
The centripetal acceleration is $4\pi^2 n^2 R$, which corresponds to option (B).