A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination 300. The co-efficient of static friction μk=0.25. If the inclination θ of the plane is increased, at what value of θ does the cylinder begin to skid, and not roll perfectly?

# A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination ${30}^{0}$. The co-efficient of static friction ${\mathrm{\mu }}_{\mathrm{k}}=0.25$. If the inclination $\mathrm{\theta }$ of the plane is increased, at what value of $\mathrm{\theta }$ does the cylinder begin to skid, and not roll perfectly?

1. A

$37°$

2. B

$53°$

3. C

$90°$

4. D

$0°$

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### Solution:

The cylinder's mass is m = 10 kg.

The cylinder's radius is r  = 15 cm (0.15 metres).

${\mathrm{\mu }}_{\mathrm{s}}$ = 0.25 is the static friction coefficient.

Angle of inclination: $\theta ={30}^{ο}$

For rolling without skidding in case of a cylinder to:

$\mathrm{\mu }\ge \frac{1}{3}\mathrm{tan\theta }$

$\mathrm{tan\theta }\le 3\mathrm{\mu }=3×0.25$

$\mathrm{\theta }\le {\mathrm{tan}}^{-1}\left(0.75\right)={37}^{°}$

Therefore for an angle above 37, the cylinder will start to skid.

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