Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 35

Calculate the force \(F(y)\) associated with each of the following potential energies: a) \(U=a y^{3}-b y^{2}\) b) \(U=U_{0} \sin (c y)\)

Short Answer

Expert verified
Question: Calculate the force, \(F(y)\), associated with the following potential energies: a) \(U = a y^{3} - b y^{2}\) b) \(U = U_{0} \sin (c y)\) Answer: a) \(F(y) = -3 a y^{2} + 2 b y\) b) \(F(y) = -U_{0} c \cos (c y)\)

Step by step solution

01

Calculate Force for Potential Energy a) \(U=a y^{3}-b y^{2}\)

First, we need to find the derivative of the potential energy: \(\frac{dU}{dy}\). So, differentiate the given potential energy \(U = a y^{3} - b y^{2}\) with respect to \(y\). \(\frac{dU}{dy} = \frac{d}{dy}(a y^{3} - b y^{2}) = 3 a y^{2} - 2 b y.\) To find the force, we simply multiply this result by \(-1\): \(F(y) = -\frac{dU}{dy} = -(3 a y^{2} - 2 b y) = -3 a y^{2} + 2 b y\).
02

Calculate Force for Potential Energy b) \(U=U_{0} \sin (c y)\)

Again, we need to find the derivative of the potential energy: \(\frac{dU}{dy}\). So, differentiate the given potential energy \(U = U_{0} \sin (c y)\) with respect to \(y\). \(\frac{dU}{dy} = \frac{d}{dy}(U_{0} \sin (c y)) = U_{0} c \cos (c y).\) To find the force, we multiply this result by \(-1\): \(F(y) = -\frac{dU}{dy} = -U_{0} c \cos (c y)\). The force associated with each potential energy is: a) \(F(y) = -3 a y^{2} + 2 b y\) b) \(F(y) = -U_{0} c \cos (c y)\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

How much mechanical energy is lost to friction if a 55.0-kg skier slides down a ski slope at constant speed of \(14.4 \mathrm{~m} / \mathrm{s}\) ? The slope is \(123.5 \mathrm{~m}\) long and makes an angle of \(14.7^{\circ}\) with respect to the horizontal.

In 1896 in Waco, Texas, William George Crush, owner of the K-T (or "Katy") Railroad, parked two locomotives at opposite ends of a 6.4 -km-long track, fired them up, tied their throttles open, and then allowed them to crash head- on at full speed in front of 30,000 spectators. Hundreds of people were hurt by flying debris; several were killed. Assuming that each locomotive weighed \(1.2 \cdot 10^{6} \mathrm{~N}\) and its acceleration along the track was a constant \(0.26 \mathrm{~m} / \mathrm{s}^{2},\) what was the total kinetic energy of the two locomotives just before the collision?

A 1.00 -kg mass is suspended vertically from a spring with \(k=100 . \mathrm{N} / \mathrm{m}\) and oscillates with an amplitude of \(0.200 \mathrm{~m} .\) At the top of its oscillation, the mass is hit in such a way that it instantaneously moves down with a speed of \(1.00 \mathrm{~m} / \mathrm{s}\). Determine a) its total mechanical energy, b) how fast it is moving as it crosses the equilibrium point, and c) its new amplitude.

A mass of \(1.00 \mathrm{~kg}\) attached to a spring with a spring constant of \(100 .\) N/m oscillates horizontally on a smooth frictionless table with an amplitude of \(0.500 \mathrm{~m} .\) When the mass is \(0.250 \mathrm{~m}\) away from equilibrium, determine: a) its total mechanical energy; b) the system's potential energy and the mass's kinetic energy; c) the mass's kinetic energy when it is at the equilibrium point. d) Suppose there was friction between the mass and the table so that the amplitude was cut in half after some time. By what factor has the mass's maximum kinetic energy changed? e) By what factor has the maximum potential energy changed?

A classmate throws a \(1.0-\mathrm{kg}\) book from a height of \(1.0 \mathrm{~m}\) above the ground straight up into the air. The book reaches a maximum height of \(3.0 \mathrm{~m}\) above the ground and begins to fall back. Assume that \(1.0 \mathrm{~m}\) above the ground is the reference level for zero gravitational potential energy. Determine a) the gravitational potential energy of the book when it hits the ground. b) the velocity of the book just before hitting the ground.
See all solutions

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Read Explanation

Magnetism

Read Explanation

Particle Model of Matter

Read Explanation

Geometrical and Physical Optics

Read Explanation

Energy Physics

Read Explanation

Famous Physicists

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free