A 300 kg piano slides 4.4 m down a 30° incline and is kept from accelerating by a man who is pushing back on it parallel to the incline. The effective coefficient of kinetic friction is 0.40.
(a) Calculate the magnitude of the force exerted by the man.
(b) Calculate the work done by the man on the piano.
(c) Calculate the work done by the friction force.
(d) What is the work done by the force of gravity?
To solve this problem we need to consider the forces acting on the piano and use the principles of Newton's laws of motion and work-energy theorem. Let's go through each part of the problem step by step.
(a) Calculate the magnitude of the force exerted by the man:
The force exerted by the man is equal in magnitude and opposite in direction to the force of friction. The formula to calculate the frictional force is given by:
Frictional force (Ff) = coefficient of friction (μ) * normal force (N)
The normal force is the force exerted by the incline on the piano perpendicular to the surface. It can be calculated using the formula:
Normal force (N) = mass of the piano (m) * acceleration due to gravity (g) * cos(theta)
where theta is the angle of inclination.
In this case, the normal force can be calculated as:
N = 300 kg * 9.8 m/s² * cos(30°)
Now, we can substitute the value of the normal force into the frictional force equation:
Ff = 0.40 * (300 kg * 9.8 m/s² * cos(30°))
Therefore, the magnitude of the force exerted by the man is equal to the magnitude of the frictional force.
(b) Calculate the work done by the man on the piano:
To calculate the work done by the man, we can use the formula:
Work (W) = force (F) * distance (d) * cos(theta)
In this case, the force exerted by the man and the distance are parallel to each other and perpendicular to the incline. Therefore, the angle between the force and the distance is 0°, so cos(0°) = 1. We can substitute the known values into the equation:
W = F * d * cos(theta)
Substitute the magnitude of the force exerted by the man and the distance traveled:
W = F * 4.4 m * cos(0°)
(c) Calculate the work done by the friction force:
The work done by a constant force is given by the formula:
Work (W) = force (F) * distance (d) * cos(theta)
In this case, the frictional force acts opposite to the direction of motion, so the angle between the force and the distance is 180°, so cos(180°) = -1. We can substitute the known values into the equation:
W = F * d * cos(theta)
Substitute the magnitude of the frictional force and the distance traveled:
W = -F * 4.4 m * cos(180°)
(d) What is the work done by the force of gravity:
The work done by the force of gravity can be calculated using the formula:
Work (W) = force (F) * distance (d) * cos(theta)
In this case, the gravitational force is acting vertically downwards, and the angle between the force and the distance is 90°, so cos(90°) = 0. We can substitute the known values into the equation:
W = F * 4.4 m * cos(90°)
Since cos(90°) = 0, the work done by the force of gravity is zero.
Now you can substitute the values into the equations to calculate the magnitude of the force exerted by the man, the work done by the man, the work done by the friction force, and the work done by the force of gravity.