A 255 kg piano slides 4.3 m down a 30° incline and is kept from accelerating by a man who is pushing back on it parallel to the incline (Fig. 6-35). The effective coefficient of kinetic friction is 0.40.
(a) Calculate 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?
(e) What is the net work done on the piano?
To answer these questions, we need to use some physics principles and formulas. In this case, we will be using force, work, and energy concepts. Let's go step by step to find the answers.
(a) Calculate the force exerted by the man:
The force exerted by the man is equal to the sum of the gravitational force component parallel to the incline and the force of kinetic friction opposing the motion.
First, calculate the gravitational force component acting parallel to the incline:
F_gravity_parallel = m * g * sin(theta)
where m is the mass of the piano (255 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and theta is the angle of the incline (30°).
F_gravity_parallel = 255 kg * 9.8 m/s² * sin(30°)
Next, calculate the force of kinetic friction:
F_friction = coefficient * F_normal
where the coefficient of kinetic friction is given as 0.40 and F_normal is the normal force acting on the piano.
The normal force can be calculated as:
F_normal = m * g * cos(theta)
F_normal = 255 kg * 9.8 m/s² * cos(30°)
Now, substitute the values to find the force exerted by the man:
Force exerted by the man = F_gravity_parallel + F_friction
Force exerted by the man = (255 kg * 9.8 m/s² * sin(30°)) + (0.40 * (255 kg * 9.8 m/s² * cos(30°)))
(b) Calculate the work done by the man on the piano:
The work done by a force is given by the formula:
Work = Force * Displacement * cos(theta)
where Force is the applied force, Displacement is the linear distance the object moves (4.3 m in this case), and theta is the angle between the applied force and the displacement.
Work done by the man = Force exerted by the man * 4.3 m * cos(theta)
(c) Calculate the work done by the friction force:
The work done by friction can be calculated using the formula:
Work = Force of Friction * Displacement * cos(180°)
Since the friction force acts opposite to the displacement, the angle between the force of friction and displacement is 180°.
Work done by friction = -F_friction * 4.3 m * cos(180°)
(d) Calculate the work done by the force of gravity:
The work done by the force of gravity will be equal to the gravitational force component parallel to the incline multiplied by the displacement:
Work done by gravity = F_gravity_parallel * 4.3 m
(e) Calculate the net work done on the piano:
The net work done on an object is equal to the sum of the work done by all the forces acting on it.
Net work done = Work done by the man + Work done by the friction + Work done by gravity
Now, plug in the calculated values to find the answers.