A 4.4 kg block is pushed along the ceiling with a constant applied force of 89 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 need to analyze the forces acting on the block and apply Newton's second law of motion.
First, let's break down the forces acting on the block:
1. The weight (W) of the block acts vertically downward and can be calculated as W = m*g, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
W = (4.4 kg) * (9.8 m/s^2) = 43.12 N
2. The normal force (N) exerted by the ceiling acts perpendicularly to the surface of contact between the block and the ceiling. Since the block is on the ceiling, the normal force is equal in magnitude and opposite in direction to the weight.
N = 43.12 N (upward)
3. The applied force (F_applied) acts at an angle of 55.0° with the horizontal. We can split this force into horizontal (F_hor) and vertical (F_ver) components.
F_hor = F_applied * cosθ
F_hor = 89 N * cos(55.0°) = 50.98 N (to the right)
F_ver = F_applied * sinθ
F_ver = 89 N * sin(55.0°) = 73.53 N (upward)
4. The frictional force (f_friction) opposes the motion of the block and acts parallel to the ceiling surface. The magnitude of the frictional force can be determined using the formula f_friction = μ_k * N, where μ_k is the coefficient of kinetic friction.
Now, using Newton's second law in the horizontal direction, we can write the following equation of motion:
F_net = F_hor - f_friction = m * a
F_hor - μ_k * N = m * a
Substituting the known values:
50.98 N - μ_k * 43.12 N = 4.4 kg * 6.00 m/s^2
Simplifying the equation:
μ_k = (50.98 N - 4.4 kg * 6.00 m/s^2) / 43.12 N
μ_k = 0.109
The coefficient of kinetic friction between the block and the ceiling is approximately 0.109.