Q.A rough inclined plane is placed on a cart moving with a constant velocity $u$ on horizontal ground. A block of mass $M$ rests on the incline. Is any work done by force of friction between the block and incline? Is there then a dissipation of energy?

The key idea is the Work-Energy Theorem, applied carefully to a single force (friction) rather than the net force on the block.

1. The cart moves with constant velocity $u$, and the block does not slip on the incline, so the block itself also moves with constant horizontal velocity $u$ — in the ground frame it undergoes a horizontal displacement $u\,t$.
2. The friction between block and incline is static (no relative sliding), of magnitude $f = Mg\sin\theta$ acting up the slope, balancing the component of the block's weight along the incline.
3. A force that has a component along an object's displacement does work. Since the static friction force has a nonzero component along the block's horizontal displacement, friction does do work on the block in the ground frame — its work is not zero even though gravity does zero work (perpendicular to the horizontal motion) and the block's kinetic energy is unchanged (constant velocity), which means the normal force's work exactly cancels friction's work. …

Setting up the situation

The cart moves with constant velocity $u$, and the block rests on the incline without sliding, so the block too moves horizontally with velocity $u$. The friction between block and incline is therefore static, of magnitude $f = Mg\sin\theta$ up the slope (balancing the component of weight along the incline, angle $\theta$).

Is work done by friction?

In the ground frame the block undergoes a horizontal displacement $u\,t$. The static friction force points up the incline and so has a component along this horizontal displacement. A force with a component along the displacement does work, so:

• Yes, friction does work on the block in the ground frame. (Its work is non-zero even though the net work of all forces on the block is zero, since the block moves at constant velocity - the normal force does equal and opposite work.)

Is there dissipation of energy? …

Concept: Work Done by Static Friction vs Dissipation of Energy

This question tests the difference between "a force does work" and "a

force dissipates energy as heat" — they are not the same thing for

static friction.

Step 1: Establish that the block moves at constant velocity $u$

The cart moves at constant velocity $u$ (no acceleration), and the block

does not slip relative to the incline, so in the ground frame the block

also moves at constant horizontal velocity $u$ — undergoing a horizontal

displacement $u\,t$ in time $t$.

Step 2: Identify the friction as static, and find its magnitude

Since there is no relative sliding, friction here is static, of

magnitude $f=Mg\sin\theta$ acting up the incline, balancing the

along-incline component of the block's weight (the block is in equilibrium

in this inertial, constant-velocity frame).

Step 3: Does friction do work in the ground frame?

The friction force acts along the incline (at some angle to the horizontal),

while the block's displacement is horizontal. Since friction has a nonzero

component along that horizontal displacement (unless the incline were

perfectly flat), friction does do nonzero work on the block. Because

the block moves at constant speed, this work is exactly cancelled by equal …