   Chapter 13, Problem 14P

Chapter
Section
Textbook Problem

An object-spring system moving with simple harmonic motion has an amplitude A. (a) What is the total energy of the system in terms of k and A only? (b) Suppose at a certain instant the kinetic energy is twice the elastic potential energy. Write an equation describing this situation, using only the variables for the mass m, velocity v, spring constant k, and position x. (c) Using the results of parts (a) and (b) and the conservation of energy equation, find the positions x of the object when its kinetic energy equals twice the potential energy stored in the spring. (The answer should in terms of A only.)

(a)

To determine
The total energy of an object-spring system in terms of amplitude A and force constant k .

Explanation

The total energy of the object-spring system at any instance is given by:

E=KE+PEs (1)

• E is the total energy of the system
• KE is the kinetic energy of the system
• PEs is the elastic potential energy of the system

The kinetic energy of the system is given by:

KE=12mv2 (2)

• m is the mass of the system
• v is the velocity of the system

The elastic potential of the system is given by:

PEs=12kx2

(b)

To determine
The equation describing an instance when the kinetic energy is twice the elastic potential energy.

(c)

To determine
The position x where the kinetic energy is twice the elastic potential energy.

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