Chapter 5: Problem 9
A man pushes on a piano with mass 180 kg; it slides at constant velocity down a ramp that is inclined at 19.0\(^\circ\) above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes (a) parallel to the incline and (b) parallel to the floor.
Short Answer
Step by step solution
Analyze the forces involved
Calculate gravitational component parallel to incline
Calculate force when pushing parallel to the incline
Calculate normal force
Calculate force when pushing parallel to the floor
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inclined Planes
- Parallel component: This is the part of the force that acts directly along the plane’s surface, causing the object to slide down or move up.
- Perpendicular component: This force acts at a right angle to the surface of the incline, helping us understand the normal force, which keeps the object pressed against the plane.
Gravitational Force
- Perpendicular to the incline: This component determines the normal force, keeping the object in contact with the surface.
- Parallel to the incline: This component causes the object to slide down the ramp, inducing motion.
Constant Velocity
If the piano slides down the inclined plane at a constant velocity, the forces parallel to the incline must cancel out, keeping the speed stable. There's no acceleration, so the gravitational force component down the incline is exactly countered by the force exerted by the man pushing the piano.
This balancing act is what lets us calculate how the pushing force must be equal to the parallel gravitational component in order to maintain constant velocity. Thus, constant velocity provides insights into equilibrium on an incline and helps determine required forces to achieve this state.
Force Components
- Parallel to Incline: This component must be countered by an applied force to hold constant velocity or induce motion. In this example, it's the man's force against gravity that ensures constant speed downhill.
- Perpendicular (Normal force): This component keeps the object pressed against the incline, counteracting the perpendicular gravitational component.