When a body slides down from rest along a smooth inclined plane making an angle of $45^o$ with the horizontal, it takes time $T$. When the same body slides down from rest along a rough inclined plane making the same angle and through the same distance, it is seen to take time $pT$, where $p$ is some number greater than $1$. Calculate the coefficient of friction between the body and the rough plane.
When a body slides down from rest along a smooth inclined plane making an angle of $45^o$ with the horizontal, it takes time $T$. When the same body slides down from rest along a rough inclined plane making the same angle and through the same distance, it is seen to take time $pT$, where $p$ is some number greater than $1$. Calculate the coefficient of friction between the body and the rough plane.
When slope (incline) is smooth as show
$\mathrm{F}=m a$
$\mathrm{~F}=m g \sin \theta$
$\therefore m a=m g \sin \theta$
$a =g \sin \theta$
$\theta =45^{\circ}$
$a=g\left(\frac{1}{\sqrt{2}}\right)$
Let distance covered by body be $d$
$d=u t+\frac{1}{2} a t^{2}$
$u=0 \text { let } t=\mathrm{T}$
$d=\frac{1}{2}\left(\frac{g}{\sqrt{2}}\right) \mathrm{T}^{2}$
$d=\frac{g \mathrm{~T}^{2}}{2 \sqrt{2}}$
When body moves on rough incline,
$f =\mu \mathrm{N}$
$=\mu m g \cos \theta$
Net force,
$=\mu g \sin \theta-\mu m g \cos \theta$
$=m(g \sin \theta-\mu g \cos \theta)$
$=m g(\sin \theta-\mu \cos \theta)$
$\mathrm{F} =m a^{\prime}$
$a^{\prime} =\text { effective acceleration, }$
$m a^{\prime} =m g(\sin \theta-\mu \cos \theta)$
$a^{\prime} =g(\sin \theta-\mu \cos \theta)$
$\theta =45$
n in figure boyd slides on slope,
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