Question

A small block of mass m is placed on a long plank of mass M that is placed on a frictionless horizontal floor. The block is abruptly given a velocity u on the plank. If during sliding of the block a distance L relative to the plank, total heat loss is Q. Find acceleration of the plank during the sliding of the block.

A. $\dfrac{Q}{{mL}}$
B. $\dfrac{Q}{{ML}}$
C. $\dfrac{Q}{{\left( {m + M} \right)L}}$
D. None of these

Answer 1

When the block slides over a plank, the heat is generated. This heat can be determined by the law of conservation of energy as the kinetic energy is being converted into the thermal energy. Use Newton’s second law to express the frictional force and the deceleration by the frictional force.

Formula used:
Thermal energy, $Q = \mu mgd$,
where, $\mu$ is the coefficient of friction, m is the mass, g is the acceleration due to gravity and d is the distance travelled by the block.

Complete Step by Step Answer:
We know that when the body slides over a rough surface, the heat is generated. Thus, when the block of mass m slides over a plank, the heat is generated. We have from the law of conservation of energy; the loss in the kinetic energy of the block is used to generate the heat Q. Therefore, we can write,
$\dfrac{1}{2}m{u^2} = Q = \mu mgL$
Here, $\mu$ is the coefficient of friction, m is the mass of the block, g is the acceleration due to gravity and L is the distance travelled by the block on the rough surface.
From the above equation, the coefficient of friction becomes,
$\mu = \dfrac{Q}{{mgL}}$ …… (1)
Now, the acceleration or deceleration of the block on the plank is due to the frictional force acting on the block.
${f_r} = \mu mg = ma$
$\Rightarrow a = \mu g$
Using equation (1) in the above equation, we get,
$a = \left( {\dfrac{Q}{{mgL}}} \right)g$
$\therefore a = \dfrac{Q}{{mL}}$

So, the correct answer is option A.

Note: There is no potential energy change in the motion of the block. The total energy of the system is the addition of kinetic energy and thermal energy produced by the friction. The block will not accelerate since the friction is acting on it to restrict its motion. Therefore, you can say the block has undergone deceleration.