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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 44c
Capitolo 6, Problema 44c

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 falls freely?

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Identify the forces acting on the mass. Since the elevator is in free fall, the only force acting on the mass is gravity, which has a magnitude of \( F_g = m g \), where \( g \) is the acceleration due to gravity.
Recognize that in the frame of reference of the elevator, the normal force from the inclined plane is the only force that prevents the mass from moving through the plane. However, since the elevator is in free fall, the effective acceleration of the mass relative to the plane is influenced by the geometry of the incline.
Decompose the gravitational force \( F_g \) into components relative to the inclined plane. The component parallel to the plane is \( F_{\text{parallel}} = m g \sin(\theta) \), and the component perpendicular to the plane is \( F_{\text{perpendicular}} = m g \cos(\theta) \), where \( \theta = 38^\circ \).
Since the elevator is in free fall, the normal force becomes zero, and the mass experiences an effective acceleration relative to the plane. The acceleration relative to the plane is given by \( a_{\text{relative}} = g \sin(\theta) \).
Substitute the known values into the formula \( a_{\text{relative}} = g \sin(\theta) \) to calculate the acceleration. Use \( g = 9.8 \ \text{m/s}^2 \) and \( \theta = 38^\circ \) to find the numerical result if needed.

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Inclined Plane Dynamics

An inclined plane is a flat surface tilted at an angle to the horizontal. When an object slides down an inclined plane, its acceleration depends on the angle of inclination and the gravitational force acting on it. The component of gravitational force acting parallel to the plane causes the object to accelerate down the slope, while the perpendicular component affects the normal force.
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Percorso guidato
06:59
Intro to Inclined Planes

Free Fall

Free fall occurs when an object is falling under the influence of gravity alone, with no other forces acting on it. In this scenario, the elevator is falling freely, meaning it is accelerating downwards at the same rate as the gravitational acceleration (approximately 9.81 m/s²). This creates a condition of weightlessness for objects inside the elevator, affecting their relative motion.
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Percorso guidato
08:36
Vertical Motion & Free Fall

Relative Acceleration

Relative acceleration refers to the acceleration of an object as observed from another object. In this case, the mass m sliding on the inclined plane experiences a different acceleration relative to the elevator due to the free fall condition. Since both the mass and the elevator are accelerating downwards at the same rate, the effective acceleration of the mass relative to the inclined plane must be calculated considering this frame of reference.
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Percorso guidato
04:27
Intro to Relative Motion (Relative Velocity)
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