A block of mass m is attached with a spring of force constant k. The coefficient of friction between the surace and the block of mass m is 3/4. Find the minimum value of mass M in terms of m so that the block of mass m starts moving up the surface.

To determine the minimum value of mass M required for the block of mass m to start moving up the surface, we need to analyze the forces acting on the block and apply Newton's laws of motion. The key here is to consider both the gravitational force acting on the block and the frictional force opposing its motion.

Understanding the Forces at Play

First, let's identify the forces acting on the block of mass m:

• Weight of the block (W): This is given by W = mg, where g is the acceleration due to gravity.
• Normal force (N): This is the force exerted by the surface on the block, acting perpendicular to the surface.
• Frictional force (F_f): This opposes the motion and is calculated as F_f = μN, where μ is the coefficient of friction.
• Force exerted by the spring (F_s): The spring exerts a force that can be expressed as F_s = kx, where x is the displacement from the spring's equilibrium position.

Setting Up the Equation

For the block to start moving up the surface, the net force acting on it must be greater than or equal to zero. The forces acting on the block can be summarized as follows:

The gravitational force pulling the block down is mg, and the frictional force opposing the motion is F_f = μN. The normal force N can be expressed as the weight of the mass M, which is N = Mg when M is hanging vertically. Therefore, the frictional force becomes:

F_f = μMg

Now, for the block to start moving up, the force exerted by the spring must overcome both the weight of the block and the frictional force:

F_s ≥ mg + F_f

kx ≥ mg + μMg

Finding the Minimum Mass M

Rearranging the equation gives us:

kx - mg ≥ μMg

Now, we can isolate M:

kx - mg ≥ (3/4)Mg

Rearranging further, we get:

kx - mg ≥ (3/4)Mg

Now, factor out M:

M ≤ (4/k)(kx - mg)/3g

To find the minimum value of M, we need to ensure that the left side is positive, which means:

kx > mg

Thus, the minimum mass M can be expressed as:

M ≥ (4/k)(kx - mg)/3g

Conclusion

In summary, the minimum value of mass M in terms of m, ensuring that the block of mass m starts moving up the surface, can be derived from the forces acting on the system. By carefully analyzing the gravitational, normal, frictional, and spring forces, we arrive at a clear expression for M. This approach not only highlights the importance of understanding forces in motion but also illustrates how to manipulate equations to solve for unknowns in physics.