91Ó°ÊÓ

We wills start with writing equilibrium equations for X and Y axes.

For Y-axis: $mg\cos \mathrm{θ}=N   ⋯⋯\left(1\right)$

We can’t write equilibrium equations for X axis, because it is not static.

Therefore, the Newton’s Second Law is given by,

$ma=mg\sin \mathrm{θ}-F    ⋯⋯\left(2\right)$, where,

$F={\mathrm{μ}}_{s}mg\cos \mathrm{θ}$, where ${\mathrm{μ}}_s$is the coefficient of static friction.

The Newton’s Second Law for rotational motion is,

$FR=\frac{2}{3}m{R}^2\mathrm{Î\pm }   ⋯⋯\left(3\right)$,

where $\mathrm{Î\pm }=\frac{a}{R}$, is an angular acceleration and R is the radius.

Thus, equation $\left(3\right)$becomes,

$$\begin{gathered} FR=\frac{2}{3}m{R}^2\frac{a}{R}    \\ ⇒F=\frac{2}{3}ma   ⋯⋯\left(4\right) \end{gathered}$$

Using this in $\left(2\right)$, we get,

$$\begin{gathered} ma=mg\sin \mathrm{θ}-\frac{2}{3}ma \\ ⇒a=g\sin \mathrm{θ}-\frac{2}{3}a \\ ⇒\left(1+\frac{2}{3}\right)a=g\sin \mathrm{θ} \end{gathered}$$

$⇒a=\frac{3}{5}g\sin \mathrm{θ}$

$$\begin{gathered} ⇒\mathrm{a}=\frac{3}{5}\left(9.80\right)\sin 38° \\ ⇒\mathrm{a}=\frac{3}{5}\left(9.80\right)\left(0.62\right) \\ ⇒\mathrm{a}=3.65,\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$

Hence, the acceleration is,$\mathrm{a}=3.65\mathrm{m}/{\mathrm{s}}^2$.

Substituting this value in $\left(4\right)$, we get,

$$\begin{gathered} F=\frac{2}{3}\left(2.00\right)\left(3.65\right) \\ F=4.86 N \end{gathered}$$

Hence, the friction force is, $F=4.86 N$.

Since this is without slipping, we have to justify this. In order to find the minimum coefficient of friction that is need to prevent slipping, we have,

$$\begin{gathered} F=\mathrm{μ}N \\ F=\mathrm{μ}mg\cos \mathrm{θ} \\ ⇒\mathrm{μ}=\frac{F}{mg\cos \mathrm{θ}} \\ ⇒\mathrm{μ}=\frac{4.86}{2\left(9.80\right)\left(0.79\right)} \\ ⇒\mathrm{μ}=0.31 \end{gathered}$$

The coefficient of frictions is, $\mathrm{μ}=0.31$.

If the mass would be doubled, the acceleration a wouldn’t be affected since from part (a) it doesn’t depends on mass. But the friction force will be doubled as it depends on mass directly. Doubling of mass wouldn’t change the minimum coefficient of friction because it doesn’t depend on mass.