Ch. 07 - Work and Energy

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. 07 - Work and Energy

Problem 52

Chapter 7, Problem 52

A 2800-kg space vehicle, initially at rest, falls vertically from a height of 2900 km above the Earth’s surface. Determine how much work is done by the force of gravity in bringing the vehicle to the Earth’s surface.

Verified step by step guidance

Determine the gravitational potential energy at the initial height (2900 km above the Earth's surface) using the formula:

U = - \frac{G M m}{r}

, where

G

is the gravitational constant,

M

is the mass of the Earth,

m

is the mass of the space vehicle, and

r

is the distance from the center of the Earth to the vehicle (Earth's radius + 2900 km).

Calculate the gravitational potential energy at the Earth's surface using the same formula:

U = - \frac{G M m}{r}

, where

r

is now the Earth's radius.

Find the work done by gravity by calculating the change in gravitational potential energy:

W = U i - U f

, where

U i

is the initial potential energy and

U f

is the final potential energy.

Substitute the known values for the gravitational constant

G

, Earth's mass

M

, Earth's radius, and the mass of the space vehicle into the equations to compute the initial and final potential energies.

Subtract the final potential energy from the initial potential energy to determine the total work done by gravity. Ensure that the units are consistent throughout the calculation (e.g., meters for distance, kilograms for mass).

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Key Concepts

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

Work Done by Gravity

Work done by a force is defined as the product of the force and the distance over which it acts, in the direction of the force. In the case of gravity, the work done can be calculated using the formula W = F * d, where F is the gravitational force and d is the distance fallen. For an object falling under gravity, this work is equal to the change in gravitational potential energy.

Gravitational Potential Energy

Gravitational potential energy (U) is the energy an object possesses due to its position in a gravitational field, calculated as U = m * g * h, where m is mass, g is the acceleration due to gravity, and h is the height above a reference point. As the vehicle falls, its potential energy decreases, and this energy is converted into kinetic energy and work done by gravity.

Acceleration Due to Gravity

The acceleration due to gravity (g) is the rate at which an object accelerates towards the Earth when in free fall, approximately 9.81 m/s² near the Earth's surface. However, this value decreases with altitude. For large heights, such as 2900 km, the average gravitational acceleration must be considered, which can be calculated using the formula g = G * M / r², where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth.

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Textbook Question

The net force exerted on a particle acts in the positive x direction. Its magnitude increases linearly from zero at x = 0, to 380 N at x = 3.0m. It remains constant at 380 N from x = 3.0m to x = 7.0m, and then decreases linearly to zero at x = 12.0m. Determine the work done to move the particle from x = 0 to x = 12.0m graphically, by determining the area under the Fₓ versus x graph.

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