Ch. 06 - Gravitation and Newton's Synthesis

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 06 - Gravitation and Newton's Synthesis

Problem 44c

Chapter 6, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Textbook Question

Use the binomial expansion ${(1 \pm x)}^{n} = 1 \pm n x + \frac{n (n - 1)}{2} x^{2} \pm \dots$ to show that the value of g is altered by approximately $Δ g \approx - 2 g \frac{Δ r}{r_{E}}$ at a height ∆r above the Earth’s surface, where r_E is the radius of the Earth, as long as ∆r ≪ r_E.

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Textbook Question

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 moves upward at constant speed?

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Textbook Question

The value of g is altered by approximately $$\Delta$ g$\thickapprox$-2g$\frac{\Delta r}{r_{E}$}$ at a height ∆r above the Earth’s surface, where r_E is the radius of the Earth, as long as ∆r ≪ r_E. What is the meaning of the minus sign in this relation?

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Textbook Question

Determine the mean distance from Jupiter for each of Jupiter’s principal moons, using Kepler’s third law. Use the mean distance of Io and the periods given in Table 6–3. Compare your results to the values in Table 6–3.

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Textbook Question

A satellite circles a spherical planet of unknown mass in a circular orbit of radius 1.6 x 10⁷ m. The magnitude of the gravitational force exerted on the satellite by the planet is 120 N. What would be the magnitude of the gravitational force exerted on the satellite by the planet if the radius of the orbit were increased to 3.0 x 10⁷m?