A man pushes on a piano with mass $180$ kg; it slides at constant velocity down a ramp that is inclined at $19.0°$ above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes parallel to the floor.

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Identify the forces acting on the piano: The piano is sliding at constant velocity, which means the net force acting on it is zero. The forces include the gravitational force (weight), the normal force from the ramp, and the force applied by the man.

Break the gravitational force into components: The weight of the piano is given by $ F_g = m \cdot g $, where $ m = 180 \ \text{kg} $ and $ g = 9.8 \ \text{m/s}^2 $. The component of the gravitational force parallel to the ramp is $ F_{g, \text{parallel}} = F_g \cdot \sin(\theta) $, and the component perpendicular to the ramp is $ F_{g, \text{perpendicular}} = F_g \cdot \cos(\theta) $, where $ \theta = 19.0^\circ $.

Determine the force applied by the man: Since the piano slides at constant velocity, the force applied by the man must counteract the parallel component of the gravitational force. However, the man is pushing parallel to the floor, so his force must be resolved into components. The horizontal force applied by the man, $ F_{\text{man}} $, has a component along the ramp given by $ F_{\text{man, ramp}} = F_{\text{man}} \cdot \cos(\theta) $.

Set up the equilibrium condition: For the piano to slide at constant velocity, the force applied by the man along the ramp must equal the parallel component of the gravitational force. Therefore, $ F_{\text{man}} \cdot \cos(\theta) = F_{g, \text{parallel}} $. Substitute $ F_{g, \text{parallel}} = F_g \cdot \sin(\theta) $ into this equation.

Solve for the magnitude of the man's force: Rearrange the equation to isolate $ F_{\text{man}} $: $ F_{\text{man}} = \frac{F_g \cdot \sin(\theta)}{\cos(\theta)} $. Substitute $ F_g = m \cdot g $ and the given values for $ m $, $ g $, and $ \theta $ to calculate $ F_{\text{man}} $.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Newton's First Law of Motion

Newton's First Law states that an object at rest will remain at rest, and an object in motion will remain in motion at a constant velocity unless acted upon by a net external force. In this scenario, the piano slides down the ramp at constant velocity, indicating that the net force acting on it is zero, which is crucial for understanding the balance of forces involved.

Components of Forces

When analyzing forces acting on an object on an inclined plane, it is essential to resolve forces into their components. The gravitational force acting on the piano can be broken down into two components: one parallel to the ramp (causing it to slide down) and one perpendicular to the ramp (acting against the normal force). This decomposition helps in calculating the net forces acting on the piano.

Force of Gravity

The force of gravity acting on an object is given by the equation F = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.81 m/s²). For the piano, with a mass of 180 kg, this force must be considered when calculating the force the man applies, especially since it influences the component of gravitational force acting parallel to the ramp.