A block of mass 3 kg slides down an inclined plane which...

A block of mass $3 \mathrm{~kg}$ slides down an inclined plane which makes an angle of $30^{\circ}$ with the horizontal. The coefficient of kinetic friction between the block and surface is $\frac{1}{2 \sqrt{3}}$. The forces that must be applied to the block for it (i) to move down the plane without any acceleration (ii) to move up the plane without any acceleration are respectively $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$ (a) $7.5 \mathrm{~N}, 22.5 \mathrm{~N}$ (b) $40 \mathrm{~N}, 25 \mathrm{~N}$ (c) $20 \mathrm{~N}, 50 \mathrm{~N}$ (d) $15 \mathrm{~N}, 21 \mathrm{~N}$

A block of mass $3 \mathrm{~kg}$ slides down an inclined plane which makes an angle of $30^{\circ}$ with the horizontal. The coefficient of kinetic friction between the block and surface is $\frac{1}{2 \sqrt{3}}$. The forces that must be applied to the block for it
(i) to move down the plane without any acceleration
(ii) to move up the plane without any acceleration are respectively $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$
(a) $7.5 \mathrm{~N}, 22.5 \mathrm{~N}$
(b) $40 \mathrm{~N}, 25 \mathrm{~N}$
(c) $20 \mathrm{~N}, 50 \mathrm{~N}$
(d) $15 \mathrm{~N}, 21 \mathrm{~N}$

Key Concepts

Inclined Plane Mechanics

This concept involves analyzing forces on objects on an inclined plane by resolving the gravitational force into components parallel and perpendicular to the surface. The parallel component (mg sin ?) tends to cause the object to slide, while the perpendicular component (mg cos ?) influences the normal force, which is crucial for calculating friction.

Frictional Forces

Friction is a resistive force that opposes the motion of an object sliding over a surface. It is calculated by multiplying the normal force by the coefficient of friction. In kinetic scenarios, this coefficient is known as the coefficient of kinetic friction and plays a vital role in determining the net force acting on the object along the plane.

Equilibrium Conditions

When an object moves without acceleration (constant velocity), the net force acting on it is zero. This concept is used to set up force balance equations, where the applied force must exactly cancel the sum of forces due to gravity and friction, whether the object moves up or down the plane.

Transcript

00:01 So in this question we have a block of mass m equals to 3 kg which is sliding down an inclined plane with angle theta equals to 30 degree the coefficient of kinetic friction mu k that is equals to 1 by 2 under root 3 okay and force that must be applied for 8 so for part 1 to move down the block okay so we can draw the free body diagram for this situation so this is the inclination plane with angle theta and this is the block and we have to apply a force f as this and this will be mg here and it will be mg sine theta and as the block is moving downward so acceleration will be downward so the friction force will be in this direction and which will be mg cos theta okay so now we can write the equation of newton's law so we can write that mg sine theta minus mu mg hos theta minus f that is equals to mass multiplied by acceleration but we have to move the plane move the block without any acceleration so this acceleration become zero so this net force this become zero okay so from here we get f that is equals to m g sine theta minus new m g cost theta so from here we can write that m g sine theta minus mu cost theta so substituting values so we get m which is three kg so three multiplied by g which is ten and sign 30 degree minus mu which is 1 by 2 under 3 and cost 30 degree okay so from here after solving we get force for this condition that is equals to 7 .5 newton in the upward direction okay so this become the answer for the part first now moving to the second part in which we have to apply this force so that the block can move up without any acceleration okay so again drawing the free body diagram for this situation so this is inclination and and this is the block and we have to apply a force so that block can move upward.

02:02 So this will be the force and this will be the acceleration and this is mg.

02:08 So the component of this mg will be here mg sine theta and as the block is moving upward, so the friction force will be in the backward direction.

02:17 So this will be the friction force.

02:18 So suppose this is mu mg cost theta.

02:21 So we can write the equation for the newton's law.

02:24 So we can write that f minus mu m.

02:28 M.

02:28 G.

02:28 Cos theta minus m g, sine theta, that is equals to mass multiplied by acceleration.

02:36 And here acceleration is also zero.

02:38 So this becomes equal to zero.

02:40 So from here f comes out to be mg, mg can be taken out...