A 4.00 kg block is pushed along the ceiling with a constant applied force of 85.0 N that acts at an angle of 55.0° with the horizontal, as in the figure below. The block accelerates to the right at 6.00 m/s2. Determine the coefficient of kinetic friction between block and ceiling.

Well, this situation seems "ceiling"-bratory! Let's solve it using some "block"-ing physics humor, shall we?

To find the coefficient of kinetic friction between the block and the ceiling, we need to first find the normal force acting on the block. Since the block is pushed along the ceiling, the normal force is equal in magnitude and opposite in direction to the force of gravity.

Using the equation F_net = m * a, where F_net represents the net force acting on the block, m is the mass of the block, and a is the acceleration, we have:

F_net = m * a

Since the net force is the vector sum of the applied force (F_applied) and the force of friction (F_friction), we can write:

F_net = F_applied - F_friction

Now, let's break down the forces:

The horizontal component of the applied force is given by F_applied_horizontal = F_applied * cos(θ), where θ is the angle between the applied force and the horizontal.

The vertical component of the applied force is given by F_applied_vertical = F_applied * sin(θ).

Now, we know that the force of friction is given by F_friction = μ * N, where μ is the coefficient of kinetic friction and N is the normal force.

Since the block is moving rightward, the frictional force opposes the motion. Therefore, the frictional force acts horizontally and is equal in magnitude to the horizontal component of the applied force.

Now, let's find the normal force (N). Since the block is on the ceiling, it experiences an upward force of magnitude equal to its weight.

The weight of the block (F_weight) is given by F_weight = m * g, where g is the acceleration due to gravity.

Since F_weight is equal in magnitude and opposite in direction to the normal force (N), we can write:

F_weight = N

So, F_net = F_applied_horizontal - F_friction

m * a = F_applied_horizontal - F_friction

m * a = F_applied * cos(θ) - μ * N

Substituting F_weight for N and rearranging the equation, we have:

m * a = F_applied * cos(θ) - μ * F_weight

Now, we can solve for the coefficient of kinetic friction (μ):

μ = (F_applied * cos(θ) - m * a) / F_weight

Plugging in the given values:

μ = (85.0 N * cos(55.0°) - 4.00 kg * 6.00 m/s²) / (4.00 kg * 9.8 m/s²)

Calculating this, we find the coefficient of kinetic friction (μ) to be approximately 0.47.

So, the coefficient of kinetic friction between the block and the ceiling is "0.47". Don't "friction" about it!

To determine the coefficient of kinetic friction between the block and the ceiling, we need to apply Newton's second law of motion.

1. Start by resolving the applied force into its horizontal and vertical components:
F_horizontal = F_applied * cos(angle)
F_vertical = F_applied * sin(angle)

Given: F_applied = 85.0 N, angle = 55.0°

F_horizontal = 85.0 N * cos(55.0°) = 85.0 N * 0.5736 ≈ 48.81 N
F_vertical = 85.0 N * sin(55.0°) = 85.0 N * 0.8192 ≈ 69.53 N

2. The horizontal component of the applied force is responsible for overcoming friction and causing acceleration. Therefore, set up an equation using Newton's second law:

F_horizontal - friction force = mass * acceleration

Let's solve for the friction force.

friction force = F_horizontal - mass * acceleration
= 48.81 N - (4.00 kg * 6.00 m/s^2)
= 48.81 N - 24.00 N
= 24.81 N

3. The friction force can also be calculated using the formula:

friction force = coefficient of friction * normal force

However, in this case, the normal force is equal to the vertical component of the applied force (F_vertical). So we can rewrite the equation as:

coefficient of friction * F_vertical = friction force

Let's solve for the coefficient of kinetic friction.

coefficient of kinetic friction = friction force / F_vertical
= 24.81 N / 69.53 N
≈ 0.356

Therefore, the coefficient of kinetic friction between the block and the ceiling is approximately 0.356.

To find the coefficient of kinetic friction between the block and the ceiling, we need to analyze the forces acting on the block.

First, let's resolve the applied force into its horizontal and vertical components.

The horizontal component of the applied force can be found using trigonometry:
F_horizontal = F_applied * cos(angle)
F_horizontal = 85.0 N * cos(55.0°)
F_horizontal = 85.0 N * 0.5736
F_horizontal = 48.756 N

Since the block is accelerating to the right, the net force in the horizontal direction can be calculated using Newton's second law:
Net Force_horizontal = mass * acceleration
48.756 N - Frictional Force = 4.00 kg * 6.00 m/s²

Now, let's determine the frictional force.
The frictional force can be calculated using the formula:
Frictional Force = coefficient of kinetic friction * normal force

The normal force is the force exerted by the ceiling on the block in the vertical direction. Since the block is pushed against the ceiling, the normal force equals the weight of the block.

Weight = mass * gravitational acceleration
Weight = 4.00 kg * 9.8 m/s²
Weight = 39.2 N

Now, substitute the known values into our equation for the frictional force:
48.756 N - frictional force = 39.2 N

Rearrange the equation to solve for the frictional force:
frictional force = 48.756 N - 39.2 N
frictional force = 9.556 N

Now, we can substitute the frictional force into the equation for the net force:
9.556 N = 4.00 kg * 6.00 m/s²

Finally, rearrange the equation to solve for the coefficient of kinetic friction:
coefficient of kinetic friction = (9.556 N) / (4.00 kg * 6.00 m/s²)

Using a calculator, the coefficient of kinetic friction is approximately 0.399.

Therefore, the coefficient of kinetic friction between the block and the ceiling is 0.399.

y direction:

85sin55 = mg + Fn (note normal is DOWN in this case) Find Fn
x direction:
85cos55-Ff = ma Find Ff
mu = Ff/Fn