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 find the answers to these questions, we can break down the problem into smaller parts and apply some fundamental principles of physics.
First, let's calculate the force of gravity acting on the piano. The force of gravity can be calculated using the formula:
F_gravity = m * g
where m is the mass of the piano and g is the acceleration due to gravity (approximately 9.8 m/s^2). In this case, the mass of the piano is given as 300 kg, so:
F_gravity = 300 kg * 9.8 m/s^2 = 2940 N
Now, let's calculate the normal force acting on the piano. The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force will act perpendicular to the incline. The normal force can be calculated using the formula:
N = m * g * cos(theta)
where theta is the angle of the incline (30°). Plugging in the values:
N = 300 kg * 9.8 m/s^2 * cos(30°) = 2550 N
(a) To find the magnitude of the force exerted by the man, we need to calculate the net force acting on the piano. The net force is the vector sum of all the forces acting on an object. Since the piano is not accelerating, the net force must be zero. We can break down the forces acting on the piano:
1. Force of gravity (F_gravity) = 2940 N (acting downward)
2. Normal force (N) = 2550 N (acting perpendicular to the incline)
3. Force exerted by the man (F_man) = ? (acting parallel to the incline)
4. Friction force (F_friction) = μ * N (where μ is the coefficient of kinetic friction)
Since there is no acceleration, the force exerted by the man must exactly balance the force of gravity and the friction force:
F_man - F_gravity - F_friction = 0
Substituting the known values:
F_man - 2940 N - (0.40 * 2550 N) = 0
F_man - 2940 N - 1020 N = 0
F_man = 3960 N
So, the magnitude of the force exerted by the man is 3960 N.
(b) To calculate the work done by the man on the piano, we use the formula:
Work = Force * Distance * cos(theta)
In this case, the force exerted by the man (F_man) is 3960 N and the distance moved by the piano (d) is given as 4.4 m. However, since the force exerted by the man is parallel to the incline, the angle (theta) between the force and the direction of motion is 0 degrees, and cos(0) = 1.
Work = 3960 N * 4.4 m * cos(0) = 17424 J
So, the work done by the man on the piano is 17424 Joules.
(c) To calculate the work done by the friction force, we use the same formula:
Work = Force * Distance * cos(theta)
In this case, the force of friction (F_friction) is given as μ * N, where μ is the coefficient of kinetic friction (0.40) and N is the normal force (2550 N). Similarly, the distance moved by the piano is 4.4 m, and since the friction force acts parallel to the incline, the angle (theta) is also 0 degrees.
Work = (0.40 * 2550 N) * 4.4 m * cos(0) = 4488 J
So, the work done by the friction force is 4488 Joules.
(d) The work done by the force of gravity can be calculated using the formula:
Work = Force * Distance * cos(theta)
In this case, the force of gravity (F_gravity) is acting downward, and the distance moved by the piano is 4.4 m. The angle (theta) between the force of gravity and the direction of motion is 180 degrees, and cos(180) = -1.
Work = (-2940 N) * 4.4 m * cos(180) = -12936 J
Note that the negative sign indicates that the force of gravity is acting in the opposite direction to the motion of the piano.
So, the work done by the force of gravity is -12936 Joules.
To solve this problem, we need to break it down into several steps:
Step 1: Calculate the component of the weight parallel to the incline
Step 2: Calculate the friction force
Step 3: Calculate the net force
Step 4: Calculate the magnitude of the force exerted by the man
Step 5: Calculate the work done by the man on the piano
Step 6: Calculate the work done by the friction force
Step 7: Calculate the work done by the force of gravity
Let's solve it step by step.
Step 1: Calculate the component of the weight parallel to the incline
The weight of the piano is given by the formula:
Weight = mass * gravity, where mass = 300 kg and gravity = 9.8 m/s^2.
Weight = 300 kg * 9.8 m/s^2 = 2940 N.
The component of weight parallel to the incline is given by:
Weight_parallel = Weight * sin(angle), where angle = 30 degrees.
Weight_parallel = 2940 N * sin(30 degrees) = 1470 N.
Step 2: Calculate the friction force
The friction force is given by the formula:
Friction = coefficient * Weight_normal, where coefficient = 0.40 and Weight_normal = Weight * cos(angle).
Weight_normal = Weight * cos(30 degrees) = 2940 N * cos(30 degrees) = 2545.05 N.
Friction = 0.40 * 2545.05 N = 1018.02 N.
Step 3: Calculate the net force
The net force is given by the formula:
Net force = Weight_parallel - Friction.
Net force = 1470 N - 1018.02 N = 451.98 N.
Step 4: Calculate the magnitude of the force exerted by the man
Since the man is pushing back on the piano parallel to the incline and keeping it from accelerating, the magnitude of the force exerted by the man is equal to the net force.
Magnitude of the force exerted by the man = 451.98 N.
Step 5: Calculate the work done by the man on the piano
The work done by a force is given by the formula:
Work = force * distance.
In this case, the force exerted by the man is parallel to the distance traveled by the piano.
Distance = 4.4 m.
Work done by the man = 451.98 N * 4.4 m = 1988.792 N·m.
Step 6: Calculate the work done by the friction force
The work done by the friction force is given by the formula:
Work = force * distance.
Since the friction force is opposite to the motion, the distance will be the same as before, but with opposite sign.
Work done by the friction force = - 1018.02 N * 4.4 m = - 4471.288 N·m.
(Note: Negative sign indicates that the work is done against the motion.)
Step 7: Calculate the work done by the force of gravity
The work done by the force of gravity is given by the formula:
Work = force * distance.
In this case, the force of gravity is acting perpendicular to the distance traveled by the piano, so the work done by the force of gravity is zero.
Therefore, the answers to the given questions are:
(a) The magnitude of the force exerted by the man is 451.98 N.
(b) The work done by the man on the piano is 1988.792 N·m.
(c) The work done by the friction force is - 4471.288 N·m.
(d) The work done by the force of gravity is zero.