oscillations in the spring-mass system maximum speed, maximum kinetic energy maximum force, maximum...

Oscillations in the spring-mass system

Maximum speed, maximum kinetic energy

Maximum force, maximum acceleration

Will the period change if the amplitude changes?

period

• Frequency and period are not dependend on the amplitude!

• What are the implications for musical instruments?

• Pitch is not dependent on how much the string is plucked (for example)

Does the period depend on the mass?

Which force is acting on both masses?

F kx

Which acceleration does each mass experience?

ma kxkx

am

The smaller mass accelerates at a higher rate: it is faster to move through the cycle.

It will oscillate at a smaller period (faster).

Properties of the spring and period

stiff soft

Large spring constant k1

Small spring constant k2

kxa

m

Large k means high acceleration

High acceleration means a cycle is completed fast

High k <-> low period

Factors influencing the period:

Influence Period Frequency

Mass m

Stiffness k

Amplitude A No influence No influence

Frequency and Period

1

2

kf

m

2m

k

Simple harmonic motion

What is influenced by the amplitude?

21

2E KE EPE kA

What happens if energy gets lost due to friction?

• A) the frequency decreases

• B) the period decreases

• C) the amplitude decreases

• D) all of the above

The amplitude decreases.

Frequency and period stay the same.

Damped harmonic motion

Example

Clink of a coffee cup

Example

• Finger flicking Short, hard to assign pitch

Driven oscillator

1spring and mass:

2

kf

m

Slow driving frequency:f0 < f

Mass moves in rhythmwith the driver.

Driven oscillator

1spring and mass:

2

kf

m

Fast driving frequency: f0 > f

The mass is practicallynot moving.

Driven oscillator

1spring and mass:

2

kf

m

Resonance:

f0 = f

Amplitude grows very much!

Amplitude versus driving frequency

Resulting amplitude

Driving frequencyNatural frequencyof oscillator

The shocks of an automobile are springs. Which of the following would

be desirable?• A) resonance with the bumps on the road,

and little damping

• B) high damping and high spring constant

• C) high damping and very low spring constant

• D) resonance with the bumps on the road, and high damping

A) resonance with the bumps on the road, and little damping

B) high damping and high spring constant

One bump, high spring constant, medium damping

One bump, high damping, medium spring constant

Several bumps, high damping, medium spring constant

Answer C: similar effects

D) resonance with the bumps on the road, and high damping

Multiple bumps

Examples of other oscillators

• Pendulum (small amplitude)

1

2

gf

L

L

Examples of other oscillators• Air-filled piston

1 1.4

2

Af

mL

m

L

A

AtmosphericPressure P0

P0

m

AtmosphericPressure P0

P0+p

m

AtmosphericPressure P0

P0-p

F=pA

Examples of other oscillators• Helmholtz Resonator

1

2

Af

VLL

V

A