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Ch. 06 - Gravitation and Newton's Synthesis
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 06 - Gravitation and Newton's SynthesisProblema 44b
Capitolo 6, Problema 44b

An inclined plane, fixed to the inside of an elevator, makes a 38° angle with the floor. A mass m slides on the plane without friction. What is its acceleration relative to the plane if the elevator accelerates downward at 0.50 g?

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Step 1: Begin by analyzing the forces acting on the mass relative to the inclined plane. The forces include the gravitational force acting vertically downward and the pseudo-force due to the downward acceleration of the elevator.
Step 2: Resolve the gravitational force into two components: one parallel to the inclined plane and one perpendicular to it. The parallel component is given by \( F_{g, \text{parallel}} = m g \sin(\theta) \), and the perpendicular component is \( F_{g, \text{perpendicular}} = m g \cos(\theta) \), where \( \theta = 38° \).
Step 3: Account for the pseudo-force caused by the downward acceleration of the elevator. This pseudo-force acts upward relative to the inclined plane and has a magnitude of \( F_{\text{pseudo}} = m a_{\text{elevator}} \), where \( a_{\text{elevator}} = 0.50 g \). Resolve this pseudo-force into components parallel and perpendicular to the inclined plane.
Step 4: Combine the forces parallel to the inclined plane. The net force parallel to the plane is \( F_{\text{net, parallel}} = m g \sin(\theta) - m a_{\text{elevator}} \sin(\theta) \). Use Newton's second law \( F = m a \) to find the acceleration relative to the plane: \( a_{\text{relative}} = g \sin(\theta) - a_{\text{elevator}} \sin(\theta) \).
Step 5: Substitute the given values into the equation. Use \( g = 9.8 \, \text{m/s}^2 \), \( a_{\text{elevator}} = 0.50 g \), and \( \theta = 38° \) to calculate the acceleration relative to the plane. Ensure the units are consistent throughout the calculation.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Inclined Plane Dynamics

An inclined plane is a flat surface tilted at an angle, which affects the forces acting on an object sliding down it. The gravitational force acting on the mass can be resolved into two components: one perpendicular to the plane, which affects the normal force, and one parallel to the plane, which causes the mass to accelerate down the slope.
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Intro to Inclined Planes

Relative Acceleration

Relative acceleration refers to the acceleration of an object as observed from a specific frame of reference. In this scenario, the acceleration of the mass must be calculated relative to the inclined plane, taking into account the downward acceleration of the elevator, which modifies the effective gravitational force acting on the mass.
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Intro to Relative Motion (Relative Velocity)

Newton's Second Law

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. This principle is essential for calculating the acceleration of the mass on the inclined plane, as it allows us to sum the forces acting on the mass and determine its resulting motion.
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Intro to Forces & Newton's Second Law
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