Question

In non-uniform circular motion, the ratio of tangential acceleration to radial acceleration is \( (r = \) radius of circle, \( V = \) speed and \( \alpha = \) angular acceleration)

• A. \( \frac{r}{V} \)
• B. \( \left( \frac{r}{V} \right)^2 \propto \)
• C. \( \left( \frac{V}{r} \right)^2 \propto \)
• D. \( \left( \frac{V}{r} \right) \propto \)

Hint:

In non-uniform circular motion, the ratio of tangential acceleration to radial acceleration depends on the square of the ratio of radius to speed.

Answer 1

The Correct Option is B

Step 1: Understanding tangential and radial acceleration.
In non-uniform circular motion, the total acceleration can be split into two components: tangential acceleration (\( a_t \)) and radial (centripetal) acceleration (\( a_r \)). - The tangential acceleration \( a_t \) is given by: \[ a_t = r \alpha \] where \( r \) is the radius and \( \alpha \) is the angular acceleration. - The radial acceleration \( a_r \) is given by: \[ a_r = \frac{V^2}{r} \] where \( V \) is the speed. Step 2: Calculate the ratio of tangential to radial acceleration.
The ratio of tangential acceleration to radial acceleration is: \[ \frac{a_t}{a_r} = \frac{r \alpha}{\frac{V^2}{r}} = \frac{r^2 \alpha}{V^2} \] Step 3: Conclusion.
Thus, the ratio of tangential acceleration to radial acceleration is proportional to \( \left( \frac{r}{V} \right)^2 \), which corresponds to option (B).