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In a non - uniform circular motaion the ...

In a non - uniform circular motaion the ratio of tangential to radial acceleration is (where, r= radius of circle, v= speed of the particle, `alpha=` angular acceleration)

A

`(alpha^2r^2)/u`

B

`(alpha^2r)/u^2`

C

`(alphar^2)/u^2`

D

`u^2/(r^2alpha)`

Text Solution

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The correct Answer is:
To solve the problem of finding the ratio of tangential to radial acceleration in non-uniform circular motion, we will follow these steps: ### Step 1: Understand the Definitions In non-uniform circular motion, a particle experiences two types of acceleration: - **Tangential Acceleration (A_t)**: This is due to the change in the speed of the particle along the circular path. It is given by the formula: \[ A_t = r \cdot \alpha \] where \( r \) is the radius of the circular path and \( \alpha \) is the angular acceleration. - **Radial (Centripetal) Acceleration (A_r)**: This is due to the change in direction of the velocity vector as the particle moves along the circular path. It is given by the formula: \[ A_r = \frac{v^2}{r} \] where \( v \) is the linear speed of the particle. ### Step 2: Write the Ratio We want to find the ratio of tangential acceleration to radial acceleration: \[ \text{Ratio} = \frac{A_t}{A_r} \] ### Step 3: Substitute the Formulas Substituting the expressions for \( A_t \) and \( A_r \) into the ratio gives: \[ \text{Ratio} = \frac{r \cdot \alpha}{\frac{v^2}{r}} \] ### Step 4: Simplify the Expression Now, simplify the ratio: \[ \text{Ratio} = \frac{r \cdot \alpha \cdot r}{v^2} = \frac{r^2 \cdot \alpha}{v^2} \] ### Step 5: Final Result Thus, the final expression for the ratio of tangential to radial acceleration is: \[ \frac{A_t}{A_r} = \frac{r^2 \cdot \alpha}{v^2} \] ### Conclusion The ratio of tangential to radial acceleration in non-uniform circular motion is given by: \[ \frac{A_t}{A_r} = \frac{r^2 \cdot \alpha}{v^2} \] ---
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