An old record player of $10 c m$ radius turns at $10 r a d / s$ while mounted on a $37^{\circ}$ incline as shown in the figure. A particle of mass m can be placed anywhere on the rotating record. If the least possible coefficient of friction that must exist for no slipping to occur is $μ$ , then find $4 μ$ .

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Solution

Draw a $F B D$ of given situation.

Find least possible coefficient of friction.

Given,
Radius $R = 10 c m$

$ω = 10 r a d / s e c$

Angle of inclination, $θ = 37^{\circ}$

According to given condition,

$f \geq m g sin θ + m r ω^{2}$

$μ m g cos θ \geq m g sin θ + m r ω^{2}$

$μ \geq tan θ + \frac{ω^{2} r}{g cos θ}$

Substitute the value.

$μ \geq tan 37^{\circ} + \frac{(10)^{2} 0.1}{10 \times \frac{4}{5}}$

$4 μ \geq 8$

Final Answer : $8$

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