Starting from rest, a 10.7 kg block slides 2.60 m down to the bottom of a frictionless ramp inclined 30.0° from the floor. The block then slides an additional 4.70 m along the floor before coming to a stop.
(a) Determine the speed of the block at the bottom of the ramp.
m/s
(b) Determine the coefficient of kinetic friction between block and floor.
(c) Determine the mechanical energy lost due to friction.
J
To determine the answers to these questions, we can analyze the motion of the block using the principles of conservation of energy and the equations of motion.
(a) To find the speed of the block at the bottom of the ramp, we can use the principle of conservation of energy. The initial potential energy of the block at the top of the ramp is equal to the sum of its final kinetic energy at the bottom of the ramp and the work done by gravity.
The potential energy at the top of the ramp is given by:
Potential energy = m * g * h
where m is the mass of the block (10.7 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height of the ramp (h = 2.60 m * sin(30°)).
The final kinetic energy at the bottom of the ramp is given by:
Kinetic energy = (1/2) * m * v^2
where v is the velocity of the block at the bottom of the ramp.
Equating the potential energy to the kinetic energy and solving for v gives:
m * g * h = (1/2) * m * v^2
v^2 = 2 * g * h
v = sqrt(2 * g * h)
Plugging in the given values:
v = sqrt(2 * 9.8 m/s^2 * 2.60 m * sin(30°))
v ≈ 7.05 m/s
Therefore, the speed of the block at the bottom of the ramp is approximately 7.05 m/s.
(b) To find the coefficient of kinetic friction between the block and the floor, we can use the fact that the block comes to a stop after sliding 4.70 m along the floor. The work done against friction is equal to the change in mechanical energy.
The work done against friction is given by:
Work against friction = force of friction * distance
where force of friction = coefficient of kinetic friction * normal force.
The normal force on the block is equal to its weight:
Normal force = m * g
where m is the mass of the block (10.7 kg), and g is the acceleration due to gravity (9.8 m/s^2).
The change in mechanical energy is equal to the initial kinetic energy at the bottom of the ramp:
Change in mechanical energy = (1/2) * m * v^2
where v is the velocity of the block at the bottom of the ramp (from part a).
Equating the work against friction to the change in mechanical energy and solving for the coefficient of kinetic friction gives:
Coefficient of kinetic friction = (Work against friction) / (normal force * distance) = (Change in mechanical energy) / (normal force * distance)
Plugging in the given values:
Coefficient of kinetic friction = (0.5 * 10.7 kg * (7.05 m/s)^2) / (10.7 kg * 9.8 m/s^2 * 4.70 m)
Coefficient of kinetic friction ≈ 0.164
Therefore, the coefficient of kinetic friction between the block and the floor is approximately 0.164.
(c) To determine the mechanical energy lost due to friction, we can calculate the work done against friction using the formula:
Work against friction = force of friction * distance.
Since the force of friction is equal to the product of the coefficient of kinetic friction and the normal force, and the normal force is equal to the weight of the block (m * g), we can write:
Work against friction = coefficient of kinetic friction * m * g * distance.
Plugging in the given values:
Work against friction = 0.164 * 10.7 kg * 9.8 m/s^2 * 4.70 m
Work against friction ≈ 78.6 J
Therefore, the mechanical energy lost due to friction is approximately 78.6 Joules.