A force of 40 N is applied at an angle of 30oabove the horizon to a 5 kg block that is at rest on a rough horizontal surface. After the block starts to move with an acceleration of 3 m/s2, what is the coefficient of kinetic friction between the block and the horizontal surface?

How do I solve a friction problem without knowing the coefficient and vice versa?

You know friction force is equal to the horizontal pulling force (adjusted for acceleration).

40cos40=friction force-mass*acceleration
= (mg-40sin30)mu-ma
solve for mu

I understand that friction WOULD equal to the horizontal pulling force IF the object is moving at constant velocity, but the question asked about friction force when there is acceleration.

Are you saying that whether or not there is acceleration, the friction force would always equal to that of the pulling force?

The friction force is opposite to the direction of motion and acceleration in this case. As Bob has noted, the friction force is reduced because the vertical component of the applied force reduces the force on the ground.

40 cos30 - friction force = mass*acceleration
40 cos30 - (mg -40 sin30)*mu = m a

mu = (40 cos30 -ma)/(mg -40 sin30)

I have a sign error in my equality.

40cos40=friction force+mass*acceleration

<<Are you saying that whether or not there is acceleration, the friction force would always equal to that of the pulling force?>>

No. That is not what we are saying. The imbalance between the pulling force and the friction force provides the scceleration, in a kinetic friction situation, as here. That is reflected in the equations that Bob and I have written.

In a static-friction situation, the pulling and friction forces are equal, until enough force is applied to cause motion.

To solve a friction problem without knowing the coefficient of friction, you need to make use of Newton's second law of motion and the concept of equilibrium. Here's how you can solve such a problem:

1. Set up the coordinate system: Choose a coordinate system where the horizontal direction is parallel to the rough surface, and the vertical direction is perpendicular to the surface (normal to gravity).

2. Determine the forces acting on the block: In this case, the applied force of 40 N can be decomposed into two components: one in the horizontal direction (F⊥) and one in the vertical direction (F∥). The applied force can be written as F = F∥ + F⊥, where F⊥ = F × sin(θ) and F∥ = F × cos(θ), and θ is the angle of the applied force with respect to the horizontal direction.

3. Calculate the net force: The net force acting on the block in the horizontal direction is given by F_net = F∥ - frictional force.

4. Apply Newton's second law: Use Newton's second law, F_net = m × a, where m is the mass of the block and a is the acceleration of the block.

5. Solve for the coefficient of kinetic friction: Since the block is moving, you can use the equation for frictional force, F_friction = μ_k × N, where μ_k is the coefficient of kinetic friction and N is the normal force acting on the block. The normal force is equal and opposite to the vertical component of the applied force, N = -F∥.

6. Substitute the equations: Substitute the expressions for F_net and F_friction into Newton's second law equation and solve for the coefficient of kinetic friction (μ_k).

Vice versa, if you know the coefficient of kinetic friction but need to find other quantities like applied force, acceleration, or net force, you can rearrange the equations in steps 4 and 6 to isolate the desired variable.

Note: In this case, since the block is already moving, you need to use the coefficient of kinetic friction rather than the coefficient of static friction.