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Ch 05: Applying Newton's Laws
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 05: Applying Newton's LawsProblema 9a
Capitolo 5, Problema 9a

A man pushes on a piano with mass 180180 kg; it slides at constant velocity down a ramp that is inclined at 19.0°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 incline.

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Identify the forces acting on the piano. Since the piano is sliding at constant velocity, the net force along the incline must be zero. This means the force applied by the man balances the component of the piano's weight acting along the incline.
Write the expression for the gravitational force acting on the piano. The weight of the piano is given by \( F_g = m \cdot g \), where \( m = 180 \; \text{kg} \) is the mass of the piano and \( g = 9.8 \; \text{m/s}^2 \) is the acceleration due to gravity.
Determine the component of the gravitational force acting along the incline. This is given by \( F_{g, \text{parallel}} = F_g \cdot \sin(\theta) \), where \( \theta = 19.0^\circ \) is the angle of the incline.
Since the piano is moving at constant velocity, the force applied by the man, \( F_{\text{man}} \), must exactly counteract \( F_{g, \text{parallel}} \). Therefore, \( F_{\text{man}} = F_g \cdot \sin(\theta) \).
Substitute the known values into the equation \( F_{\text{man}} = m \cdot g \cdot \sin(\theta) \) to calculate the magnitude of the force applied by the man. Ensure the angle \( \theta \) is converted to radians if necessary for trigonometric calculations.

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Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In this scenario, since the piano slides at constant velocity, the net force acting on it is zero, meaning the applied force by the man must balance the component of gravitational force acting down the ramp.
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Gravitational Force Component

When an object is on an inclined plane, the gravitational force acting on it can be resolved into two components: one parallel to the incline and one perpendicular to it. The parallel component, which causes the object to slide down the ramp, is calculated using the formula F_parallel = mg sin(θ), where m is the mass, g is the acceleration due to gravity, and θ is the angle of the incline.
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Constant Velocity

Constant velocity implies that an object's speed and direction remain unchanged over time. In this case, since the piano is moving down the ramp at constant velocity, the forces acting on it are balanced. This means the force exerted by the man must equal the gravitational force component acting parallel to the incline, allowing for a straightforward calculation of the required force.
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