Question

A small block of mass $m$ is kept on the rough inclined surface of inclination $\theta$ fixed in an elevator. The elevator goes up with a uniform velocity $v$ and the block does not slide on the wedge. The work done by the force of friction on the block in time t will be

Answer 1

Step 1: Given data

1. The mass of the block is $m$
2. The angle of the inclined surface is $\theta$.
3. The velocity of the elevator is $v$.

Step 2: Frictional force and work done

1. The frictional force is the force, which resists the motion of a body over another surface in contact with it.
2. The frictional force is defined by the form, $F=\mu N$, where, $\mu$ is the coefficient of friction $\left(\mu =\tan \theta \right)$ and $N$ is the normal force exerted on the body.
3. Work done is the total amount of energy to bring a body from one place to another. Work done is the product of force and displacement of a body, which is defined as, $W=F.S$, where, F is the force exerted on the body and S is the displacement.

Step 3: Finding the frictional force

From the question, the elevator goes up with a uniform velocity $v$ . That's why we taking the vertical component of the frictional force exerted on it.

The vertical displacement of the elevator is $S=vt............\left(1\right)$ $\left(Since,displacement=velocity\times time\right)$.

The vertical component of frictional force is $F_v=F\sin \theta$ and the normal force exerted on the block is $N=mg\cos \theta$ (from the figure ).

Again we know, that frictional force is $F=\mu N$

So,

$$\begin{gathered} F_v=F\sin \theta =\mu N\sin \theta \\ or{F}_v=\left(\tan \theta \right).mg.\cos \theta .\sin \theta \\ or{F}_v=\frac{\sin \theta }{\cos \theta }.mg.\cos \theta .\sin \theta \\ or{F}_v=mg.{\sin }^2\theta ................\left(2\right) \end{gathered}$$

Step 4: Finding the work done

Again, we know that work done $W=F.S$.

So,

$$\begin{gathered} W=F.S=mg.{\sin }^2\theta .vt(u\sin gequations1and2) \\ orW=mgvt.{\sin }^2\theta \end{gathered}$$

Therefore, the work done by the force of friction on the block in time t is $mg.vt.{\sin }^2\theta$.