A 370kg piano slides 2.9m down a 24∘ incline and is kept from accelerating by a man who is pushing back on it parallel to the incline. Ignore friction.
Is there a question here?
To solve this problem, we need to consider the forces acting on the piano along the incline. Since we are ignoring friction, the only forces to consider are the force of gravity acting straight downward and the force exerted by the man pushing parallel to the incline.
Let's break down the forces:
1. Force due to gravity (weight):
The weight of the piano can be calculated using the formula: weight = mass × acceleration due to gravity.
Given: mass of the piano = 370 kg and acceleration due to gravity = 9.8 m/s^2.
Therefore, weight = 370 kg × 9.8 m/s^2 = 3626 N (rounded to the nearest whole number).
2. Force exerted by the man (up the incline):
The force exerted by the man is equal in magnitude but opposite in direction to the force of gravity, as he is preventing the piano from accelerating. Therefore, the force exerted by the man is 3626 N.
Since the piano is sliding down the incline, we need to calculate the component of the weight that acts parallel to the incline.
The weight can be resolved into two components: one parallel to the incline (force down the incline) and one perpendicular to the incline (normal force). Since the piano is not moving vertically, the normal force is equal in magnitude to the perpendicular component of the weight but opposite in direction.
The parallel component of the weight (force down the incline) can be calculated using the formula:
force down the incline = weight × sin(θ)
Given: angle of the incline (θ) = 24°.
force down the incline = 3626 N × sin(24°) = 1566 N (rounded to the nearest whole number).
Therefore, the man needs to exert a force of 1566 N in the opposite direction (up the incline) to prevent the piano from accelerating.
Note: This analysis assumes ideal conditions without considering frictional forces.