Question

A block of mass M is pushed up a frictionless incline which makes an angle theta with respect to the horizontal. The pushing force is parallel to the incline and has a constant magnitude. A small block of mass m sits on top of the large block. The coefficient of static friction between the two blocks is μ. Derive an expression for the maximum magnitude P of the pushing force for which the small block does not slip on the large block, in terms of relevant system parameters. (Hint: begin with fully labeled free-body diagrams for the small block and for the two blocks combined, respectively.)

Answer 1

Step-by-step explanation:

To find the maximum magnitude P of the pushing force for which the small block does not slip on the large block, we'll start by analyzing the forces acting on the small block and the forces acting on the combined system of both blocks. We'll use the following variables:

- M: Mass of the large block

- m: Mass of the small block

- θ: Angle of the incline with respect to the horizontal

- μ: Coefficient of static friction between the two blocks

- g: Acceleration due to gravity (approximately 9.81 m/s²)

**Free-Body Diagram for the Small Block (m):**

- There is a gravitational force acting downward: mg.

- There is a normal force acting perpendicular to the incline: N (normal to the incline).

- There is a frictional force between the small block and the large block: μN.

- There is the pushing force P acting parallel to the incline.

**Free-Body Diagram for the Combined System (M + m):**

- There is a gravitational force acting downward: (M + m)g.

- There is a normal force acting perpendicular to the incline: N (normal to the incline).

- There is the pushing force P acting parallel to the incline.

Now, we'll derive the expression for the maximum P without causing the small block to slip. The key is to consider the conditions at the point where the small block is on the verge of slipping, which means the maximum frictional force is reached.

1. **For the small block (m):**

The maximum frictional force that the small block can experience without slipping is μN. So, we have:

\[μN = P \sin θ\] (since the small block doesn't move vertically, N = mg cos θ)

2. **For the combined system (M + m):**

The net force acting on the combined system parallel to the incline is equal to the mass times acceleration:

\[P - μN = (M + m)g \sin θ\]

Now, we can substitute N from the first equation into the second equation:

\[P - μ(mg \cos θ) = (M + m)g \sin θ\]

Now, we can solve for P:

\[P = (M + m)g \sin θ + μ(mg \cos θ)\]

This is the expression for the maximum magnitude P of the pushing force for which the small block does not slip on the large block.