A force of 60N acts on a box with a mass of 35.0 kg sitting on the floor. The force acts at an angle of 30º with the horizontal, and the coefficient of kinetic friction between the surfaces is 0.15.
a) What is the normal force on the box exerted by the floor?
b) What is the magnitude of the resultant acceleration of the box?
To find the answers to these questions, we need to analyze the forces acting on the box and apply Newton's laws of motion.
a) To determine the normal force on the box exerted by the floor, we need to consider the gravitational force acting on the box and the vertical component of the applied force. The normal force is equal in magnitude but opposite in direction to the gravitational force.
First, we calculate the gravitational force:
Gravitational force (Fg) = mass (m) x acceleration due to gravity (g)
Fg = (35.0 kg) x (9.8 m/s^2) ⟶ Fg = 343 N
Next, we find the vertical component of the applied force:
Vertical component of force = Applied force x sin(θ)
Vertical component of force = (60 N) x sin(30º) ⟶ Vertical component of force = 30 N
Since the normal force is equal in magnitude but opposite in direction to the gravitational force, the normal force exerted by the floor is:
Normal force = Gravitational force + Vertical component of force
Normal force = 343 N + 30 N ⟶ Normal force = 373 N
Therefore, the normal force on the box exerted by the floor is 373 N.
b) To find the magnitude of the resultant acceleration of the box, we need to consider the horizontal component of the applied force and the frictional force acting on the box.
First, we find the horizontal component of the applied force:
Horizontal component of force = Applied force x cos(θ)
Horizontal component of force = (60 N) x cos(30º) ⟶ Horizontal component of force = 51.96 N
Next, we calculate the frictional force:
Frictional force = coefficient of friction x normal force
Frictional force = (0.15) x (373 N) ⟶ Frictional force = 55.95 N
Since the applied force is greater than the frictional force, a net force in the horizontal direction is present. Therefore, we can calculate the resultant acceleration using Newton's second law:
Resultant force (in the horizontal direction) = mass x acceleration
(Resultant force) - (Frictional force) = mass x acceleration
(51.96 N) - (55.95 N) = (35.0 kg) x acceleration
Rearranging the equation to isolate the acceleration:
acceleration = (51.96 N - 55.95 N) / (35.0 kg)
acceleration = -3.99 N / (35.0 kg)
acceleration = -0.114 m/s^2
Therefore, the magnitude of the resultant acceleration of the box is 0.114 m/s^2.