A 215 kg crate is being pushed across a horizontal floor by a force \( P \) that makes an angle of 30.5° below the horizontal. The coefficient of kinetic friction is 0.265. What should be the magnitude of \( P \), so that the net work done by it and the kinetic frictional force is zero?

To find the magnitude of the force P, we need to calculate the net work done by it and the kinetic frictional force. If the net work is zero, the force P will be balanced by the kinetic frictional force.

Explanation:

To find the magnitude of the force P, we need to calculate the net work done by it and the kinetic frictional force. If the net work is zero, the force P will be balanced by the kinetic frictional force.

The net work done is given by the formula: W_net = W_P - W_fr, where W_P is the work done by force P and W_fr is the work done by kinetic friction.

Since the crate is being pushed horizontally, the work done by P can be calculated as W_P = P * d * cos(theta), where d is the displacement of the crate. The work done by kinetic friction can be calculated as W_fr = mu_k * N * d, where mu_k is the coefficient of kinetic friction and N is the normal force.

Setting the net work to zero, we can solve for P: 0 = P * d * cos(theta) - mu_k * N * d. Now, substitute the values and solve for P.