Answer:

University Physics Volume 1

15 Oscillations

15.2 Energy in Simple Harmonic Motion

LEARNING OBJECTIVES

By the end of this section, you will be able to:

Describe the energy conservation of the system of a mass and a spring

Explain the concepts of stable and unstable equilibrium points

To produce a deformation in an object, we must do work. That is, WHETHER you pluck a guitar string or COMPRESS a car’s shock absorber, a force must be exerted through a distance. If the only result is deformation, and no work goes into thermal, sound, or kinetic energy, then all the work is initially STORED in the deformed object as some FORM of potential energy.

Consider the example of a block attached to a spring on a frictionless table, oscillating in SHM. The force of the spring is a conservative force (which you studied in the chapter on potential energy and conservation of energy), and we can define a potential energy for it. This potential energy is the energy stored in the spring when the spring is extended or compressed. In this CASE, the block oscillates in one dimension with the force of the spring acting parallel to the motion:

W

=

x

f

∫

x

i

F

x

d

x

=

x

f

∫

x

i

−

k

x

d

x

=

[

−

1

2

k

x

2

]

x

f

x

i

=

−

[

1

2

k

x

2

f

−

1

2

k

x

2

i

]

=

−

[

U

f

−

U

i

]

=

−

Δ

U

.

When considering the energy stored in a spring, the equilibrium position, marked as

x

i

=

0.00

m,

is the position at which the energy stored in the spring is equal to zero. When the spring is stretched or compressed a distance x, the potential energy stored in the spring is

U

=

1

2

k

x

2

.

Energy and the Simple Harmonic Oscillator

To study the energy of a simple harmonic oscillator, we need to consider all the forms of energy. Consider the example of a block attached to a spring, placed on a frictionless surface, oscillating in SHM. The potential energy stored in the deformation of the spring is

U

=

1

2

k

x

2

.