A 5.55 kg block covered in sandpaper is pushed along the ceiling of a room under construction. The block is pushed across the ceiling with a force of 77.9 N directed at an angle of 71.5° to the horizontal. If the coefficient of kinetic friction between the paper and the ceiling is 0.574, what acceleration does the block undergo?
See:
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To find the acceleration of the block, we'll need to consider the forces acting on it and then apply Newton's second law of motion, which states that the net force on an object is equal to its mass multiplied by its acceleration (F = ma).
First, let's break down the forces acting on the block:
1. The force applied (F_applied) is the force with which the block is being pushed across the ceiling. Its magnitude is given as 77.9 N, and its direction is 71.5° to the horizontal.
2. The force of friction (F_friction) opposes the motion and is caused by the sandpaper on the ceiling. Its magnitude can be determined using the coefficient of kinetic friction (μ_k) and the normal force (N) between the block and the ceiling. The normal force is equal to the block's weight, which is given by the mass of the block (5.55 kg) multiplied by the acceleration due to gravity (9.8 m/s^2).
Now, let's calculate the magnitude of the normal force:
N = mg = (5.55 kg) * (9.8 m/s^2) = 54.39 N
Next, we can calculate the magnitude of the force of friction:
F_friction = μ_k * N = 0.574 * 54.39 N = 31.25 N
Now, let's resolve the applied force into its horizontal and vertical components:
F_applied_horizontal = F_applied * cos(71.5°)
F_applied_vertical = F_applied * sin(71.5°)
F_applied_horizontal = 77.9 N * cos(71.5°) = 77.9 N * 0.3174 = 24.76 N
F_applied_vertical = 77.9 N * sin(71.5°) = 77.9 N * 0.9488 = 73.96 N
Now, let's calculate the net force in the horizontal direction:
Net force_horizontal = F_applied_horizontal - F_friction
Net force_horizontal = 24.76 N - 31.25 N = -6.49 N
The negative sign indicates that the net force in the horizontal direction opposes the applied force.
Finally, we can use Newton's second law of motion to determine the acceleration of the block:
Net force_horizontal = mass * acceleration
-6.49 N = 5.55 kg * acceleration
Solving for acceleration:
acceleration = -6.49 N / 5.55 kg ≈ -1.17 m/s^2
The block undergoes an acceleration of approximately -1.17 m/s^2 along the ceiling. The negative sign indicates that the acceleration is in the opposite direction of the applied force and friction force.