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 44b

Chapter 6, Problem 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?

Name: Physics for Scientists and Engineers
Author: Giancoli Douglas
ISBN: 9780137488179

Verified step by step guidance

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.

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, 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.

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.

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.

Recommended video:

Guided course

06:54

Intro to Forces & Newton's Second Law

Related Practice

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.

1481

views

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?

1428

views

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?

1279

views

Textbook Question

Four masses are arranged as shown in Fig. 6–28. Determine the x and y components of the gravitational force on the mass at the origin (m). Write the force in vector notation (î, ĵ).

1838

views

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.