Question

In the shown figure, friction coefficients are shown and the surface between '2m' and block 'A' is smooth. Initially, the system is at rest with spring in natural length (spring constant = K). Now a constant horizontal force 'F is applied on the block A, so that box '2m' just starts moving. What is the minimum possible value of this 'F':

Answer 1

F = μmg (Assuming spring is connected between A and 2m)

Answer 2

This problem is ambiguous, the solution assumes the spring is connected between block A and the 2m mass, which is not clear from the problem statement.

Assumptions:

• The spring connects block A and the 2m mass.
• We seek the minimum force F to initiate the motion of the 2m mass.

Forces on the 2m mass:

• Spring force (Kx) to the right
• Friction (μmg) to the left

Condition for motion:

For the 2m mass to start moving, the spring force must equal the maximum static friction:

$Kx = \mu mg$

Forces on Block A:

• Applied force (F) to the right
• Spring force (Kx) to the left

$F - Kx = ma_A$

$F = Kx + ma_A$

Minimum Force:

To minimize F, we consider the moment the 2m mass starts moving, where its acceleration is approximately zero, and we assume $a_A$ is also approximately zero.

$F = \mu mg + ma_A$

$F_{min} = \mu mg$