Conservation of energy Suppose an object with mass m moves ina region R in a conservative force field given by F = -Vọ, whereo is a potential function in a region R. The motion of the object isgoverned by Newton's Second Law of Motion, F = ma, wherea is the acceleration. Suppose the object moves from point A topoint B in R.dva. Show that the equation of motion is m --Vọ.dt1 d2 dtc. Take the dot product of both sides of the equation in part (a)with v(t) = r'(t) and integrate along a curve between A andB. Use part (b) and the fact that F is conservative to show thatdvb. Show that -dtthe total energy (kinetic plus potential) - m|v]² + q is thesame at A and B. Conclude that because A and B are arbitrary,energy is conserved in R.

Answered: Image /qna-images/answer/c6437517-d26a-45a1-8bc5-421243fc54e5.jpg

Answered: Conservation of energy Suppose an…

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter7: Conservation Of Energy

Section: Chapter Questions

Problem 60P

See similar textbooks

Similar questions

A block slides at constant speed down a ramp while acted on by three forces: its weight, the normal force, and kinetic friction. Respond to each statement, true or false. (a) The combined net work done by all three forces on the block equals zero. (b) Each force does zero work on the block as it slides. (c) Each force does negative work on the block as it slides.
In the Back to the Future movies (https://openstaxcollege.org/l/2lbactofutclip), a DeLorean car of mass 1230 kg travels at 88 miles per hour to venture back to the future. (a) What is the kinetic energy of the DeLorean? (b) What spring constant would be needed to stop this DeLorean in a distance of 0.1m?
A roller-coaster car of mass 1.50 103 kg is initially at the top of a rise at point . It then moves 35.0 m at an angle of 50.0 below the horizontal to a lower point . (a) Find both the potential energy of the system when the car is at points and and the change in potential energy as the car moves from point to point , assuming y = 0 at point . (b) Repeat part (a), this time choosing y = 0 at point , which is another 15.0 m down the same slope from point .
Give an example of a situation in which there is a force and a displacement, but the force does no work. Explain why it does no work.
A student expends 7.5 W of power in lifting a textbook 0.50 m in 1.0 s with a constant velocity. (a) How much work is done, and (b) how much does the book weigh (in newtons)? The answers to Confidence Exercises may be found at the back of the book.
A 1.50 103-kg car starts from rest and accelerates uniformly to 18.0 m/s in 12.0 s. Assume that, air resistance remains constant at 4.00 102 N during this time. Find (a) the average power developed by the engine and (b) the instantaneous power output of the engine at t = 12.0 s, just before the car stops accelerating.
The force on a particle of mass 2.0 kg varies with position according to F(x)=3.0x2 (x in meters, F(x) in newtons). The particle’s velocity at x=2.0m is 5.0 m/s. Calculate the mechanical energy of the particle using (a) the origin as the reference point and (b) x = 4.0 m as the reference point. (c) Find the particle’s velocity at x=1.0m . Do this part of the problem for each reference point.
A 5.00-kg block is set into motion up an inclined plane with an initial speed of i = 8.00 m/s (Fig. P7.21). The block comes to rest after traveling d = 3.00 m along the plane, which is inclined at an angle of = 30.0 to the horizontal. For this motion, determine (a) the change in the blocks kinetic energy, (b) the change in the potential energy of the block-Earth system, and (c) the friction force exerted on the block (assumed to be constant), (d) What is the coefficient of kinetic friction? Figure P7.21
Consider the following scenario. A car for which friction is not negligible accelerates from rest down a hill, running out of gasoline after a short distance (see below). The driver lets the car coast farther down the hill, then up and over a small crest. He then coasts down that hill into a gas station, where he brakes to a stop and fills that tank with gasoline. Identify the forms of energy the car has, and how they are changed and transferred in this series of events.
The chin-up is one exercise that can be used to strengthen the biceps muscle. This muscle can exert a force of approximately 8.00 102 N as it contracts a distance of 7.5 cm in a 75-kg male.3 (a) How much work can the biceps muscles (one in each arm) perform in a single contraction? (b) Compare this amount of work with the energy required to lift a 75-kg person 40. cm in performing a chin-up. (c) Do you think the biceps muscle is the only muscle involved in performing a chin-up?
As a simple pendulum swings back and forth, the forces acting on the suspended object are the force of gravity, the tension in the supporting cord, and air resistance, (a) Which of these forces, if any, does no work on the pendulum? (b) Which of these forces does negative work at all times during the pendulums motion? (c) Describe the work done by the force of gravity while the pendulum is swinging.
A 2.0-kg block starts with a speed of 10 m/s at the bottom of a plane inclined at 37 to the horizontal. The coefficient of sliding friction between the block and plane is k=0.30 . (a) Use the work-energy principle to determine how far the block slides along the plane before momentarily coming to rest. (b) After stopping, the block slides back down the plane. What is its speed when It reaches the bottom? (Hint: For the round trip, only the force of friction does work on the block.)

SEE MORE QUESTIONS

Recommended textbooks for you

Principles of Physics: A Calculus-Based Text

Physics

ISBN:

9781133104261

Author:

Raymond A. Serway, John W. Jewett

Publisher:

Cengage Learning

Inquiry into Physics

Physics

ISBN:

9781337515863

Author:

Ostdiek

Publisher:

Cengage

College Physics

Physics

ISBN:

9781305952300

Author:

Raymond A. Serway, Chris Vuille

Publisher:

Cengage Learning

Principles of Physics: A Calculus-Based Text

Physics

ISBN:

9781133104261

Author:

Raymond A. Serway, John W. Jewett

Publisher:

Cengage Learning

Inquiry into Physics

Physics

ISBN:

9781337515863

Author:

Ostdiek

Publisher:

Cengage

College Physics

Physics

ISBN:

9781305952300

Author:

Raymond A. Serway, Chris Vuille

Publisher:

Cengage Learning

College Physics

Physics

ISBN:

9781285737027

Author:

Raymond A. Serway, Chris Vuille

Publisher:

Cengage Learning

University Physics Volume 1

Physics

ISBN:

9781938168277

Author:

William Moebs, Samuel J. Ling, Jeff Sanny

Publisher:

OpenStax - Rice University

College Physics

Physics

ISBN:

9781938168000

Author:

Paul Peter Urone, Roger Hinrichs

Publisher:

OpenStax College

SEE MORE TEXTBOOKS