Question

In non uniform circular motion, the ratio of radial acceleration to tangential acceleration is (V is the velocity, r is the radius and \(α\) is the angular acceleration)

• A. \(\frac{αr}{V}\)
• B. \(\frac{V^2}{r^2α}\)
• C. \(\frac{r^2α}{V^2}\)
• D. \(\frac{rα^2}{V^2}\)

Hint:

Radial acceleration is V^2/r and tangential acceleration is alpha times r.

Answer 1

The Correct Option is B

Step 1: Check dimensions of the options:
The ratio of two accelerations is dimensionless. $V^2/(r^2\alpha)$: $\dfrac{\text{m}^2\text{s}^{-2}}{\text{m}^2\text{s}^{-2}}$ works, since $\alpha$ has unit $\text{s}^{-2}$.

Step 2: Check the others:
A: $\dfrac{\alpha r}{V}$ has unit $\dfrac{\text{s}^{-2}\text{m}}{\text{m s}^{-1}}=\text{s}^{-1}$, not dimensionless. D: $\dfrac{r\alpha^2}{V^2}$ has unit $\text{s}^{-2}\text{m}^{-1}$, not dimensionless. C is the inverse of B, so it is correct only if the ratio is tangential over radial.

Step 3: Pick:
Option B, which equals $\dfrac{V^2/r}{\alpha r}$.

Final Answer:
Radial acceleration is V^2/r, tangential is alpha r, so the ratio is V^2/(r^2 alpha). \[ \boxed{\text{(B) }\dfrac{V^2}{r^2\alpha}} \]