(C) Let $L$ be the length of the incline.For a smooth plane,the acceleration is $a_1 = g \sin\theta$. The time taken is $t_1 = \sqrt{2L/a_1}$.For a ro

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(C) Let $L$ be the length of the incline.For a smooth plane,the acceleration is $a_1 = g \sin\theta$. The time taken is $t_1 = \sqrt{2L/a_1}$.For a rough plane,the acceleration is $a_2 = g(\sin\theta - \mu \cos\theta)$. The time taken is $t_2 = \sqrt{2L/a_2}$.Given that $t_2 = 1.5 t_1$,we have $t_2^2 = 2.25 t_1^2$.Substituting the expressions for $t_1$ and $t_2$,we get $\frac{2L}{a_2} = 2.25 \frac{2L}{a_1}$,which simplifies to $a_1 = 2.25 a_2$.Substituting $a_1$ and $a_2$,we get $g \sin\theta = 2.25 g(\sin\theta - \mu \cos\theta)$.Since $\theta = 45^\circ$,$\sin\theta = \cos\theta = 1/\sqrt{2}$.Dividing by $g \sin\theta$,we get $1 = 2.25(1 - \mu)$.Thus,$1 - \mu = 1/2.25 = 4/9$.Therefore,$\mu = 1 - 4/9 = 5/9$.

Class 11 Physics Newton's Laws of Motion and Friction Motion (or rest) on Rough Inclined Surface

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The time taken by a block of mass $m$ to slide down from the highest point to the lowest point on a rough inclined plane is $50\%$ more compared to the time taken by the same block on an identical inclined smooth plane. Both inclined planes are at $45^\circ$ with the horizontal. The coefficient of kinetic friction between the rough inclined surface and the block is . . . . . . .

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Class 11

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Newton's Laws of Motion and Friction

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Motion (or rest) on Rough Inclined Surface

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There are two inclined surfaces of equal length $(L)$ and the same angle of inclination $45^{\circ}$ with the horizontal. One of them is rough and the other is perfectly smooth. $A$ given body takes $2$ times as much time to slide down the rough surface than on the smooth surface. The coefficient of kinetic friction $(\mu_k)$ between the object and the rough surface is close to:

DifficultNEET 2025

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$A$ car of weight $W$ is on an inclined road that rises by $100 \, m$ over a distance of $1 \, km$ and applies a constant frictional force $\frac{W}{20}$ on the car. While moving uphill on the road at a speed of $10 \, m/s$,the car needs power $P$. If it needs power $\frac{P}{2}$ while moving downhill at speed $v$,then the value of $v$ is ........ $m/s$.

DifficultJEE MAIN 2016

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$A$ block of mass $m$ rests on a rough inclined plane. The coefficient of friction between the surface and the block is $\mu$. At what angle of inclination $\theta$ of the plane to the horizontal will the block just start to slide down the plane?

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$A$ block of mass $8\, kg$ is at rest on a rough inclined plane as shown in the figure below. The magnitude of the net force exerted by the surface on the block will be ........ $N$ $(g = 10\, m/s^2)$.

Medium

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$A$ body starts from rest on a long inclined plane of slope $45^o$. The coefficient of friction between the body and the plane varies as $\mu = 0.3x$,where $x$ is the distance travelled down the plane. The body will have maximum speed (for $g = 10 \ m/s^2$) when $x = $ ........ $m$.

MediumJEE MAIN 2013

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$A$ block of mass $m$ rests on a rough inclined plane. The coefficient of friction between the surface and the block is $\mu$. At what angle of inclination $\theta$ of the plane to the horizontal will the block just start to slide down the plane?

$A$ block of mass $8\, kg$ is at rest on a rough inclined plane as shown in the figure below. The magnitude of the net force exerted by the surface on the block will be ........ $N$ $(g = 10\, m/s^2)$.

$A$ body starts from rest on a long inclined plane of slope $45^o$. The coefficient of friction between the body and the plane varies as $\mu = 0.3x$,where $x$ is the distance travelled down the plane. The body will have maximum speed (for $g = 10 \ m/s^2$) when $x = $ ........ $m$.

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