The coefficient of static friction between a block and a horizontal floor is 0.42, while the coefficient of kinetic friction is 0.17. The mass of the block is 3.9 kg. If a horizontal force is slowly increased until it is barely enough to make the block start moving, what is the net force on the block the instant that it starts to slide

To determine the net force on the block the instant it starts to slide, we need to consider the balance of forces acting on the block.

When the block is stationary, the force of static friction between the block and the floor opposes the applied force. The maximum value of static friction can be calculated using the equation:

f_static_max = coefficient_of_static_friction * normal_force

where the normal force is equal to the weight of the block in this case, as the block is on a horizontal floor. The weight or the gravitational force is given by:

weight = mass * acceleration_due_to_gravity

Thus,

weight = 3.9 kg * 9.8 m/s^2 = 38.22 N

Now, we can calculate the maximum static friction force:

f_static_max = 0.42 * 38.22 N = 16.04 N

Therefore, as long as the applied force is less than or equal to 16.04 N, the block remains stationary.

Once the applied force exceeds the maximum static friction force, the block starts to move. At this point, the static friction force changes to kinetic friction force, which is usually lower. The equation for kinetic friction force is:

f_kinetic = coefficient_of_kinetic_friction * normal_force

where

f_kinetic = 0.17 * 38.22 N = 6.48 N

The net force is the applied force minus the kinetic friction force:

net_force = applied_force - f_kinetic

Since the applied force that begins to make the block slide is slightly greater than the maximum static friction force, the net force can be approximated as:

net_force ≈ f_static_max

Therefore, the net force on the block, the instant it starts to slide, is approximately 16.04 N.