   Chapter 13, Problem 1CQ

Chapter
Section
Textbook Problem

An object–spring system undergoes simple harmonic motion with an amplitude A. Does the total energy change if the mass is doubled but the amplitude isn’t changed? Are the kinetic and potential energies at a given point in its motion affected by the change in mass? Explain.

To determine

Whether the total energy of an object-spring system changes if the mass is doubled but the amplitude isn't changed and whether the kinetic and potential energies at a given point in its motion is affected by the change in mass.

When the mass of the object is doubled but the amplitude isn't changed, the total energy of the system will remain the same. Both the kinetic energy and the potential energy at a certain point will not get affected by change in the mass

Explanation

The total energy of an object-spring system when it is in its turning point is stored in the form of elastic potential energy.

E=12kA2

• E is the total energy.
• k is the spring constant.
• A is the amplitude of the motion.

From above equation, the total energy will not change if the amplitude of the motion remains constant provided the spring constant of the spring is same.

The total energy of the system is given by,

E=12kx2+12mv2

• x is the displacement of the object from equilibrium position
• m is the mass of the object
• v is the velocity of the object

Since the total energy remains same, the sum of kinetic energy and potential energy at any point of time will remain constant.

Conclusion:

When the mass of the object is doubled but the amplitude isn't changed, the total energy of the system will remain the same. Both the kinetic energy and the potential energy at a certain point will not get affected by change in the mass.

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