A 3.40 kg block is pushed along the ceiling with an constant applied force of F = 82.0 N that acts at an angle θ = 50° with the horizontal. The block accelerates to the right at 5.80 m/s2. Determine the coefficient of kinetic friction between the block and the ceiling.

μk =

To determine the coefficient of kinetic friction (μk) between the block and the ceiling, we can start by analyzing the forces acting on the block.

1. Vertical Forces: Since the block is pushed along the ceiling, the only vertical force acting on it is gravity (mg), where m is the mass of the block and g is the acceleration due to gravity (9.8 m/s^2). However, since the block is pushed against the ceiling and does not experience vertical motion, the vertical forces cancel each other out.

2. Horizontal Forces: There are two horizontal forces acting on the block: the applied force (F) and the force of kinetic friction (fk). The applied force is given as F = 82.0 N at an angle θ = 50° with the horizontal. We need to find the force of kinetic friction (fk).

We can start by resolving the applied force into its horizontal and vertical components. The horizontal component is Fx = F * cos(θ), and the vertical component is Fy = F * sin(θ). In this case, Fx = 82.0 N * cos(50°) and Fy = 82.0 N * sin(50°).

Since the acceleration of the block is given as 5.80 m/s^2 in the horizontal direction, we can use Newton's second law (F = ma) to find the net horizontal force acting on the block. The net force is equal to the applied force minus the force of kinetic friction (fk):

Net Force = F - fk = m * a

Substituting the given values, we have:

82.0 N * cos(50°) - fk = 3.40 kg * 5.80 m/s^2

Now, we need to determine the value of fk, the force of kinetic friction. To do this, we can use the equation:

fk = μk * N

Where μk is the coefficient of kinetic friction and N is the normal force. In this case, the normal force is equal to the weight of the block, which is given by mg.

fk = μk * mg

Substituting the known values, we have:

82.0 N * cos(50°) - μk * mg = 3.40 kg * 5.80 m/s^2

Finally, we can solve for μk:

μk = (82.0 N * cos(50°) - 3.40 kg * 5.80 m/s^2) / (m * g)

Substituting the appropriate values for m and g, you can calculate the coefficient of kinetic friction (μk) between the block and the ceiling.