A student, crazed by final exams, uses a force P of magnitude 80 N to push a 4.0 kg block across the ceiling of his room. If the coefficient of kinetic friction between the block and the ceiling's surface is 0.42, what is the magnitude of the block's acceleration?
40.1 N
To find the magnitude of the block's acceleration, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object times its acceleration.
First, let's determine the force of kinetic friction acting on the block. The force of kinetic friction can be calculated using the equation F_kinetic = μ_kinetic * N, where μ_kinetic is the coefficient of kinetic friction and N is the normal force.
Since the block is being pushed upwards against the force of gravity, the normal force is equal to the weight of the block, which is given by N = m * g, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the mass of the block is 4.0 kg, the normal force is N = 4.0 kg * 9.8 m/s^2 = 39.2 N.
Now, we can calculate the force of kinetic friction by multiplying the coefficient of kinetic friction (0.42) by the normal force: F_kinetic = 0.42 * 39.2 N = 16.464 N.
Since the force of kinetic friction acts in the opposite direction to the applied force, the net force acting on the block is P - F_kinetic = 80 N - 16.464 N = 63.536 N.
Finally, we can use Newton's second law to calculate the magnitude of the block's acceleration. Rearranging the equation, we have:
net force = mass * acceleration
63.536 N = 4.0 kg * acceleration.
Dividing both sides of the equation by 4.0 kg, we find:
acceleration = 63.536 N / 4.0 kg.
Thus, the magnitude of the block's acceleration is 15.884 m/s^2.