A man pushes on a piano with mass 200 {\rm kg} so that it slides at constant velocity down a ramp that is inclined at 10.5^\circ above the horizontal floor. Neglect any friction acting on the piano.
To determine the force with which the man pushes the piano, we need to analyze the forces acting on the piano. In this case, there are two main forces to consider:
1. The force of gravity (weight): The weight is equal to the mass of the piano multiplied by the acceleration due to gravity, which is approximately 9.8 m/s^2. The weight acts vertically downward and can be represented by the equation:
weight = mass * acceleration due to gravity
weight = 200 kg * 9.8 m/s^2
weight = 1960 N
2. The normal force: The normal force is the force exerted by the ramp on the piano perpendicular to the surface of the ramp. It counteracts the force of gravity and prevents the piano from sinking into the ramp. Since the piano is sliding down the ramp at constant velocity (without sinking into the ramp), the normal force is equal in magnitude and opposite in direction to the force of gravity.
normal force = weight (because the piano is not sinking)
normal force = 1960 N
Since there is no friction acting on the piano, the only force that needs to be considered is the force of gravity. Therefore, the man must exert an equal and opposite force to counteract the force of gravity. Thus, the force with which the man pushes the piano is also 1960 N in the opposite direction (up the ramp).
In summary, the man needs to exert a force of 1960 N in an upward direction to counteract the force of gravity and allow the piano to slide down the ramp at a constant velocity.