(a) Two boxes of different mass are at rest on a ramp. What is the minimum coefficient of static friction between the boxes and the ramp if the system is in equilibrium? Given M=(1+0.34)m

(b) Suppose there is no friction between the upper box and the ramp. What is minimum coefficient of friction between the ramp and the lower box is needed for equilibrium?

Mass doesn't matter. The angle and the friction coefficient do.

To find the minimum coefficient of static friction between the boxes and the ramp for the system to be in equilibrium, we can use the concept of forces and equilibrium.

(a) Let's analyze the forces acting on the system. We have the force of gravity acting vertically downwards on both boxes. Let's denote the mass of the upper box as M and the lower box as m. The force of gravity acting on the upper box is given by M * g, where g is the acceleration due to gravity. The force of gravity acting on the lower box is m * g.

Now, let's consider the horizontal forces. The only horizontal force acting on the system is the force of static friction between the boxes and the ramp. Let's denote the coefficient of static friction between the boxes and the ramp as μ.

For the system to be in equilibrium, the sum of the horizontal forces acting on the system must be zero. This means the static friction force between the boxes and the ramp must balance out the component of the force of gravity pulling the boxes down the ramp.

The component of the force of gravity acting down the ramp can be calculated using trigonometry. Since the system is at rest, this component should be equal to the static friction force.

The component of the force of gravity acting down the ramp is given by M * g * sin(θ), where θ is the angle between the ramp and the horizontal.

So, we have: μ * (M * g * cos(θ)) = M * g * sin(θ)

Now, from the given information, we know that M = (1 + 0.34)m. Let's substitute this into the equation above and simplify:

μ * ((1 + 0.34)m * g * cos(θ)) = (1 + 0.34)m * g * sin(θ)

Next, we can cancel out the m, g, and cos(θ) terms from both sides of the equation:

μ * (1 + 0.34) = 1 + 0.34 * sin(θ)
μ * 1.34 = 1 + 0.34 * sin(θ)

Finally, rearranging the equation to solve for μ:

μ = (1 + 0.34 * sin(θ)) / 1.34

This equation gives us the minimum coefficient of static friction between the boxes and the ramp for the system to be in equilibrium.

(b) If there is no friction between the upper box and the ramp, the force of gravity acting on the upper box will not contribute to any horizontal forces. Therefore, the minimum coefficient of friction between the ramp and the lower box needed for equilibrium will be the horizontal component of the force of gravity acting on the lower box divided by the force of gravity itself.

The horizontal component of the force of gravity acting on the lower box is given by m * g * sin(θ). The force of gravity acting on the lower box is m * g. So, the minimum coefficient of friction, μ, can be found using the equation:

μ = (m * g * sin(θ)) / (m * g)

Simplifying the equation gives:

μ = sin(θ)

Hence, the minimum coefficient of friction between the ramp and the lower box needed for equilibrium is given by sin(θ), where θ is the angle between the ramp and the horizontal.