A block of mass M1 = 0.320 kg is initially at rest on a cart of mass M2 = 0.680 kg with the cart initially at rest on a level air track. The coefficient of static friction between the block and the cart is ?s = 0.610, but there is essentially no friction between the air track and the cart. The cart is accelerated by a force of magnitude F parallel to the air track. Find the maximum value of F that allows the block to accelerate with the cart, without sliding on top of the cart.

Question

A block of mass M1 = 0.320 kg is initially at rest on a cart of mass M2 = 0.680 kg with the cart initially at rest on a level air track. The coefficient of static friction between the block and the cart is ?s = 0.610, but there is essentially no friction between the air track and the cart. The cart is accelerated by a force of magnitude F parallel to the air track. Find the maximum value of F that allows the block to accelerate with the cart, without sliding on top of the cart.

Answer 1

To find the maximum value of F that allows the block to accelerate with the cart without sliding on top of the cart, we need to consider the forces acting on the block.

First, let's analyze the forces acting on the block:

1. Weight (mg): The force due to gravity acting on the block. Its magnitude can be calculated using the equation Fg = m * g, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s²).

2. Normal force (N): The force exerted by the cart on the block, perpendicular to the surface of contact between the block and the cart. Its magnitude is equal to the weight of the block since the block is initially at rest and there is no vertical acceleration. Therefore, N = Fg.

3. Static friction force (fs): The force of static friction acting between the block and the cart. The maximum value of static friction force is given by the equation fs = μs * N, where μs is the coefficient of static friction.

Since we want to find the maximum value of F, we need to find the maximum value of static friction force that can be applied before the block starts sliding on top of the cart. This occurs when the static friction force is at its maximum, given by fs = μs * N.

Replacing N with Fg, we have:

fs = μs * Fg

Substituting the values given for the coefficients of static friction (μs = 0.610) and the masses (m = 0.320 kg), we can calculate the maximum value of static friction force:

fs = 0.610 * (0.320 kg * 9.8 m/s²)

Simplifying the expression:

fs = 0.610 * 3.136 N

Calculating the maximum value of static friction force:

fs ≈ 1.909 N

Therefore, the maximum value of the force F that allows the block to accelerate with the cart without sliding on top of the cart is approximately 1.909 N.