Question 11: Laws of Motion

Question

A body of mass $5 	imes 10^{-3} \mathrm{~kg}$ is launched upon a rough inclined plane making an angle of $30^{\circ}$ with the horizontal. Obtain the coefficient of friction between the body and the plane if the time of ascent is half of the time of descent.

Accepted Answer

According to question, the situation can be
shown
Case 1: For body projected up the plane
$\begin{aligned}
& v=0 \
& v^2=u^2-2 a_1 s \
& 	ext { i.e. } \quad 0=u^2-2 a_1 s \
& 	herefore \quad u=\sqrt{2 a_1 s} \
& 	ext { Also, } \quad v=u-a_1 t \
& 	ext { i.e. } \quad 0=u-a_1 t_1 \
& t_1=u / a_1 \
& 	herefore \quad t_1=\frac{\sqrt{2 a_1 s}}{a_1}=\sqrt{\frac{2 s}{a_1}} \quad 	ext{...... (i)}\
\end{aligned}$
Retardation
$a_1=\frac{\mu R+m g \sin 30^{\circ}}{m}$
$\begin{aligned}
	ext{As} \quad R & =m g \cos 30^{\circ} \
a_1 & =\frac{\mu m g \cos 30^{\circ}+m g \sin 30^{\circ}}{m} \
& =\mu g \frac{\sqrt{3}}{2}+\frac{g}{2} \quad 	ext{..... (ii)}
\end{aligned}$
Case 2: For body coming down the plane
$\begin{aligned}
& s=u t+\frac{1}{2} a_2 t_2^2 \
& u=0 \
& t_2=\sqrt{\frac{2 s}{a_2}} \quad 	ext{..... (iii)}
\end{aligned}$
Downward acceleration
$a_2=\frac{m g \sin 30^{\circ}-\mu R}{m}$
$\begin{aligned}
& =\frac{m g \sin 30^{\circ}-\mu m g \cos 30^{\circ}}{m} \
& =\frac{g}{2}-\frac{\mu g \sqrt{3}}{2}
\end{aligned}$
Given, $t_1=\frac{t_2}{2}$
Substituting values of t$_1$ and t$_2$ from Eqs. (i) and (iii), we get
$\sqrt{\frac{2 s}{a_1}}=\frac{1}{2} \sqrt{\frac{2 s}{a_2}}$
Now, squaring, both sides we get
$a_1=4 a_2$ ..... (iv)
Substituting values of $a_1$ and $a_2$ from Eqs. (ii) and (iv), we get
$\mu g \frac{\sqrt{3}}{2}+\frac{g}{2}=4\left(\frac{g}{2}-\frac{\mu g \sqrt{3}}{2}ight)$
Solving for $\mu$, we get
$\begin{aligned}
\mu & =\frac{\sqrt{3}}{5}=\frac{1.732}{5} \
& =0.346
\end{aligned}$

Answer

0.346

Answer

0.921

Answer

1.926

Answer

2.912

Laws of Motion

A body of mass 5 \times 10^{-3} \mathrm{~kg} is launched upon a rough inclined plane making an angle of 30^{\circ} with the horizontal. Obtain the coefficient of friction between the body and the plane if the time of ascent is half of the time of descent.

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