The ratio of the accelerations for a solid sphere (mass ‘m’ and radius ‘R’) rolling down an incline of angle ’θ’ without slipping and slipping down the incline without rolling is

# The ratio of the accelerations for a solid sphere (mass ‘m’ and radius ‘R’) rolling down an incline of angle $’\mathrm{\theta }’$ without slipping and slipping down the incline without rolling is

1. A

2:3

2. B

2:5

3. C

7:5

4. D

5:7

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### Solution:

The solid sphere's roll down on an incline without slipping will have an acceleration of,

${\mathrm{a}}_{\mathrm{roll}}=\frac{\mathrm{gsin\theta }}{1+\frac{{\mathrm{K}}^{2}}{{\mathrm{r}}^{2}}}$

For solid sphere,

${\mathrm{K}}^{2}=\frac{2}{5}{\mathrm{R}}^{2}⇒\frac{{\mathrm{K}}^{2}}{{\mathrm{r}}^{2}}=\frac{2}{5}$

$⇒{\mathrm{a}}_{\mathrm{roll}}=\frac{\mathrm{gsin\theta }}{1+\frac{2}{5}}=\frac{5}{7}\mathrm{gsin\theta ______}\left(1\right)$

When a solid sphere slides down an incline without rolling, it accelerates due to,

${\mathrm{a}}_{\mathrm{slip}}=\mathrm{gsin\theta _____}\left(2\right)$

from equation 1 and 2

$\frac{{\mathrm{a}}_{\mathrm{roll}}}{{\mathrm{a}}_{\mathrm{slip}}}=\frac{5}{7}⇒{\mathrm{a}}_{\mathrm{roll}}:{\mathrm{a}}_{\mathrm{slip}}=5:7.$

Hence the correct answer is 5:7.

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