A 275 kg piano slides 4.0 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.
a. Wt.=m*g = 275kg * 9.8N/kg=2695 N =
Wt of piano.
Fp = 2695*sin30 = 1348 N. = Force parallel to incline.
Fn = 2695*cos30 = 2334 N. = Normal = Force perpendicular to incline
Fk = u*Fn = 0.40 * 2334 = 933.6 N. =
Force of kinetic friction.
Fex-Fp-Fk = m*a
Fex-1348-933.6 = m*0 = 0
Fex - 2282 = 0
Fex = 2282 N. Exerted by man.
c. Work = Fk*d = 933.6 * 4 = 3734 J.
Correction: Fex-Fp+Fk = M*a.
Repeat all calculations.
To solve this problem, we can break it down into several steps:
Step 1: Draw a diagram and identify the forces acting on the piano.
a) The forces acting on the piano are its weight (mg) pointing downward, the normal force (N) perpendicular to the incline, the force exerted by the man (F_man) pushing back parallel to the incline, and the friction force (f) opposing the motion.
Step 2: Calculate the weight of the piano (mg).
The weight of the piano can be calculated using the formula:
Weight = mass x gravitational acceleration
Given that the mass of the piano (m) is 275 kg and the gravitational acceleration (g) is approximately 9.8 m/s^2, we can calculate the weight:
Weight = 275 kg x 9.8 m/s^2 = 2695 N
Step 3: Calculate the normal force (N).
The normal force is equal to the component of the weight perpendicular to the incline, which can be found using trigonometry.
Normal force (N) = weight x cos(theta)
Where theta is the angle of the incline. In this case, theta is given as 30 degrees, so we have:
N = 2695 N x cos(30°) = 2333 N
Step 4: Calculate the friction force (f).
The friction force can be calculated using the formula:
Friction force = coefficient of friction x normal force
Given that the coefficient of kinetic friction (μ) is 0.40 and the normal force (N) is 2333 N, we can calculate the friction force:
Friction force = 0.40 x 2333 N = 933.2 N
Step 5: Calculate the force exerted by the man (F_man).
Since the piano is kept from accelerating, the force exerted by the man is equal to the parallel component of the weight plus the friction force.
F_man = weight x sin(theta) + friction force
F_man = 2695 N x sin(30°) + 933.2 N
= 1347.5 N + 933.2 N
= 2280.7 N
Therefore, the magnitude of the force exerted by the man is 2280.7 N.
Step 6: Calculate the work done by the man on the piano.
The work done by the man can be calculated using the formula:
Work = force x distance x cos(theta)
Given that the force exerted by the man (F_man) is 2280.7 N and the distance traveled by the piano (d) is 4.0 m, we can calculate the work done by the man:
Work = 2280.7 N x 4.0 m x cos(30°)
= 4561.4 J
Therefore, the work done by the man on the piano is 4561.4 Joules.
Step 7: Calculate the work done by the friction force.
The work done by the friction force can be calculated using the formula:
Work = force x distance x cos(theta),
where the force is the friction force and the distance traveled by the piano is 4.0 m.
Work = 933.2 N x 4.0 m x cos(180°)
= -3732.8 J
The negative sign indicates that the work done by the friction force is opposing the motion.
Therefore, the work done by the friction force is -3732.8 Joules.
To summarize:
a) The magnitude of the force exerted by the man is 2280.7 N.
b) The work done by the man on the piano is 4561.4 Joules.
c) The work done by the friction force is -3732.8 Joules.