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Question

# Why is the following situation impossible? A book sits on an inclined plane on the surface of the Earth. The angle of the plane with the horizontal is $60.0^{\circ}$. The coefficient of kinetic friction between the book and the plane is 0.300. At time t = 0, the book is released from rest. The book then slides through a distance of 1.00 m, measured along the plane, in a time interval of 0.483 s.

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We know that the angle of inclination is $\theta=60.0\text{\textdegree}$, the coefficient of kinetic friction between book surface, the inclination surface is $\mu=0.300$ and the acceleration due to gravity is $g=9.81\ {\rm m/s^{2}}$. The book was released with initial velocity $u=0$ at time $t=0$ and traveled a distance $d=1.00\ {\rm m}$. We have to calculate the the time taken to travel the distance. If the calculated time does not matches with the given time interval $t=0.483\ {\rm s}$ then the given situation is impossible.

Consider the mass of the book is $m$, hence, the weight of the book is $w=mg$. The weight of the book act vertically downwards. Now the component of the weight parallel to the surface of inclination is $w_{\parallel}=w\sin\theta=mg\sin\theta$.

Now, the component of the weight perpendicular to the surface of inclination is $w_{\perp}=w\cos\theta=mg\cos\theta$. Hence, the normal reaction of the surface on the book is $N=mg\cos\theta$. Hence, the friction force is $F_{f}=\mu N=\mu mg\cos\theta$.

Hence, the total force acting on the books is

\begin{align*} F & =w_{\parallel}-F_{f}\\ & =mg\sin\theta-\mu mg\cos\theta \end{align*}

Hence, the acceleration of the book is

\begin{align*} a & =\frac{F}{m}\\ & =\frac{mg\sin\theta-\mu mg\cos\theta}{m}\\ & =g\sin\theta-\mu g\cos\theta \end{align*}

Now substituting the values we have

\begin{align*} a & =\left(9.81\ {\rm m/s^{2}}\right)\sin\left(60.0\text{\textdegree}\right)-\left(0.300\right)\left(9.81\ {\rm m/s^{2}}\right)\cos\left(60.0\text{\textdegree}\right)\\ & =7.02\ {\rm m/s^{2}} \end{align*}

Hence, time taken by the book to travel the distance $d=1.00\ {\rm m}$ is

\begin{align*} t & =\sqrt{\frac{2d}{a}}\\ & =\sqrt{\frac{2\left(1.00\ {\rm m}\right)}{\left(7.02\ {\rm m/s^{2}}\right)}}\\ & =0.534\ {\rm s} \end{align*}

Hence, the book can not reach the distance in 0.483 second, hence, the given situation is impossible.

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