Question

A small block B of mass m is placed on a plank P of mass M and length l, which is placed on a frictionless horizontal floor. Coefficient of friction between the block and the plank is $\mu$. Initially the block is near the right edge of the plank and the setup is at rest. Such a constant horizontal force F is applied on the plank that the block begins to slide on the plank. Find work done by the force F, work done by the frictional force on the block, and work done by the fictional force on the plank, when the block reaches the left edge of the plank.

Answer 1

Work done by the force F: $W_F = \frac{Fl(F - \mu mg)}{F - \mu g(m+M)}$

Work done by the frictional force on the block: $W_{f,B} = \frac{\mu^2 m g^2 l M}{F - \mu g(m+M)}$

Work done by the frictional force on the plank: $W_{f,P} = -\frac{\mu mg l (F - \mu mg)}{F - \mu g(m+M)}$

Answer 2

1. Setup and Forces:

• Block (mass $m$) and Plank (mass $M$, length $l$).
• Coefficient of kinetic friction between block and plank: $\mu$.
• Plank on frictionless floor.
• Constant horizontal force $F$ applied to the plank.
• Block slides on the plank.

2. Accelerations:

• Friction force on block by plank (to the right): $f_k = \mu mg$.
• Acceleration of block B: $a_B = \frac{f_k}{m} = \frac{\mu mg}{m} = \mu g$. (to the right)
• Friction force on plank by block (to the left): $f_k = \mu mg$.
• Acceleration of plank P: $a_P = \frac{F - f_k}{M} = \frac{F - \mu mg}{M}$. (to the right)

3. Relative Displacement and Time:

• The block moves from the right edge to the left edge of the plank, meaning its relative displacement with respect to the plank is $l$.
• Relative acceleration: $a_{rel} = a_P - a_B = \frac{F - \mu mg}{M} - \mu g = \frac{F - \mu mg - M\mu g}{M} = \frac{F - \mu g(m+M)}{M}$.
• Using $l = \frac{1}{2} a_{rel} t^2$, the time taken is $t^2 = \frac{2l}{a_{rel}} = \frac{2lM}{F - \mu g(m+M)}$.

4. Displacements of Block and Plank:

• Displacement of block B: $x_B = \frac{1}{2} a_B t^2 = \frac{1}{2} (\mu g) \left(\frac{2lM}{F - \mu g(m+M)}\right) = \frac{\mu g l M}{F - \mu g(m+M)}$.
• Displacement of plank P: $x_P = \frac{1}{2} a_P t^2 = \frac{1}{2} \left(\frac{F - \mu mg}{M}\right) \left(\frac{2lM}{F - \mu g(m+M)}\right) = \frac{(F - \mu mg)l}{F - \mu g(m+M)}$.

5. Work Done by Various Forces:

• Work done by force F ($W_F$): $W_F = F \cdot x_P = F \left(\frac{(F - \mu mg)l}{F - \mu g(m+M)}\right) = \frac{Fl(F - \mu mg)}{F - \mu g(m+M)}$.
• Work done by frictional force on the block ($W_{f,B}$): The friction force on the block is $f_k = \mu mg$ (to the right), and the block's displacement $x_B$ is also to the right. $W_{f,B} = f_k \cdot x_B = (\mu mg) \left(\frac{\mu g l M}{F - \mu g(m+M)}\right) = \frac{\mu^2 m g^2 l M}{F - \mu g(m+M)}$.
• Work done by frictional force on the plank ($W_{f,P}$): The friction force on the plank is $f_k = \mu mg$ (to the left), while the plank's displacement $x_P$ is to the right. $W_{f,P} = -f_k \cdot x_P = -(\mu mg) \left(\frac{(F - \mu mg)l}{F - \mu g(m+M)}\right) = -\frac{\mu mg l (F - \mu mg)}{F - \mu g(m+M)}$.