A 4.8 kg block is pushed along the ceiling with a constant applied force of 87 N that acts at an angle of 55.0° with the horizontal, as in Figure 4-33. The block accelerates to the right at 6.00 m/s2. Determine the coefficient of kinetic friction between the block and the ceiling.
To determine the coefficient of kinetic friction between the block and the ceiling, we can use Newton's second law of motion and consider the forces acting on the block.
First, let's consider the forces acting on the block:
1. The applied force F_app acting at an angle of 55.0° with the horizontal.
2. The force of gravity acting vertically downward, which can be calculated as F_gravity = m * g, where m is the mass of the block (4.8 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
3. The normal force N exerted by the ceiling on the block, which is the force perpendicular to the contact surface. Since the block is moving along the ceiling, the normal force is equal in magnitude but opposite in direction to the force of gravity, N = m * g.
4. The force of kinetic friction F_friction, which acts in the opposite direction of motion.
Using Newton's second law of motion, we can write the equation of motion in the horizontal direction:
ΣF_horizontal = m * a
F_app * cos(55.0°) - F_friction = m * a
Here, F_app * cos(55.0°) is the horizontal component of the applied force.
Now, let's solve for the force of kinetic friction:
F_friction = F_app * cos(55.0°) - m * a
Substituting the given values:
F_friction = 87 N * cos(55.0°) - 4.8 kg * 6.00 m/s^2
Calculating this, we find:
F_friction ≈ 50.82 N
Finally, we can use the formula for kinetic friction to find the coefficient:
F_friction = μ_k * N
μ_k = F_friction / N
Since N = m * g, we can substitute the expression for N:
μ_k = F_friction / (m * g)
μ_k = 50.82 N / (4.8 kg * 9.8 m/s^2)
Calculating this, we find:
μ_k ≈ 1.08
Therefore, the coefficient of kinetic friction between the block and the ceiling is approximately 1.08.
To determine the coefficient of kinetic friction, we can use Newton's second law of motion, which states that the net force on an object is equal to the mass of the object multiplied by its acceleration.
First, we need to resolve the applied force into its horizontal and vertical components. The horizontal component is given by F_applied * cos θ, where θ is the angle of the applied force.
F_applied_horizontal = 87 N * cos 55°
F_applied_horizontal = 87 N * 0.5736
F_applied_horizontal = 49.901 N
The only force acting horizontally on the block is the force of kinetic friction, which opposes the applied force. So, the net force (F_net) is equal to the force of kinetic friction (F_friction).
F_net = F_friction
Using Newton's second law, we can write:
F_net = m * a
Substituting the values:
F_friction = m * a
F_friction = 4.8 kg * 6.00 m/s^2
F_friction = 28.8 N
Now, we can find the coefficient of kinetic friction using the formula:
F_friction = μ_k * N
where N is the normal force. The normal force is equal to the weight of the block, since the block is pushed against the ceiling.
Normal force (N) = m * g
N = 4.8 kg * 9.8 m/s^2
N = 47.04 N
Now, we can solve for the coefficient of kinetic friction (μ_k):
F_friction = μ_k * N
28.8 N = μ_k * 47.04 N
Dividing both sides by 47.04 N:
μ_k = 28.8 N / 47.04 N
μ_k = 0.613
Therefore, the coefficient of kinetic friction between the block and the ceiling is approximately 0.613.