Physics 221 march 29
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Key Concepts: The pendulum Damped and driven oscillation Traveling waves Standing waves. Physics 221, March 29. The simple pendulum. Hooke’s law: F = - kx  x(t) = x max cos ( ω t + φ)) Masses on springs obey Hooke’s law and their motion is simple harmonic motion. The pendulum:

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Physics 221, March 29

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Physics 221 march 29

  • Key Concepts:

  • The pendulum

  • Damped and driven oscillation

  • Traveling waves

  • Standing waves

Physics 221, March 29


The simple pendulum

The simple pendulum

Hooke’s law: F = -kx x(t) = xmaxcos(ωt + φ)) Masses on springs obey Hooke’s law and their motion is simple harmonic motion.

The pendulum:

For a given displacement from equilibrium s = Lθ, the net force on the pendulum bob is F =-mgsinθ.

For small displacements sinθ ~ θ, so we can write F =-mgθ = - (mg/L)*Lθ = -ks, with k = (mg/L)

F obeys Hooke’s law.Hooke’s law  simple harmonic motion

s(t) = smaxcos(ωt + φ)) ω2 = k/m = g/L

, f = 1/T


Physics 221 march 29

Extra credit:A simple pendulum consists of a point mass m suspended by a massless, unstretchable string of length L. If the mass is doubled while the length of the string remains the same, the period of the pendulum

  • becomes 4 times greater.

  • becomes twice as great.

  • becomes greater by a factor of square root of 2.

  • remains unchanged.

  • decreases.

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Physics 221 march 29

A simple pendulum of length L oscillates back and forth 10 times per second. By what factor do you have to change its length to make it oscillate back and forth only 5 times per second.

  • Increase the length by a factor of √2.

  • Increase the length by a factor of 2.

  • Increase the length by a factor of 4.

  • Decrease the length by a factor of √2.

  • Decrease the length by a factor of 2.

  • Decrease the length by a factor of 4.

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Physics 221 march 29

A simple pendulum of length L oscillates back and forth 10 times per second.

By what factor do you have to change its length to make it oscillate back and

forth only 5 times per second.

  • Original frequency:10/s

  • Original period:(1/10)s

  • Desired frequency:5/s

  • Desired period:(1/5)s

    You want to double the period.

    You must increase L by a factor of 4.

    Link: Pendulum Waves


Damped oscillations

Damped oscillations

Free oscillations of macroscopic systems damp out.

If the total force on the damped system is

F = -kx – bv,

then the motion of the system is given by

x(t) = A0exp(-bt/2m)cos(ωdampt + φ),

with ωdamp2 = k/m - (b/2m)2.

The energy stored in the system is proportional to the square of its amplitude, it decreases as E = E0exp(bt/m).

A -> A0/2 implies E -> E0/4


Physics 221 march 29

You are watching a mass oscillate on a spring. The initial amplitude is 10 cm. You measure the period to be a constant 1.1 s. The energy E stored in the oscillating system halves after 10 s. How long does it take for the amplitude of the oscillation to decrease to 5 cm?

  • 5 s

  • 10 s

  • 20 s

  • 40 s

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Forced oscillations

Forced oscillations

Assume we have a damped harmonic oscillator acted on by a driving force F0cos ωextt .

The net force acting on the oscillator is

F = F0cosωextt - kx – bv.

The resulting motion is

x(t) = Acos(ωextt + φ).

The amplitude of forced oscillations is small unless the driving frequency

is close to the natural frequency.

When the driving frequency is equal to the natural frequency the amplitude of

the oscillations can be become very large - this is called resonance.

http://www.acoustics.salford.ac.uk/feschools/waves/wine3video.htm


Physics 221 march 29

Consider a tractor driving across a field that has undulations at regular intervals. The distance between the bumps is about 4.2 m. Because of safety reasons, the tractor does not have a suspension system but the driver’s seat is attached to a spring to absorb some of the shock as the tractor moves over rough ground. The natural frequency of the driver and the seat is f = 2 Hz. At what speed does the drive become dangerous?

  • 2.1 m/s = 4.7 mph

  • 4.2 m/s = 9.4 mph

  • 8.4 m/s = 18.8 mph

  • 12.6 m/s = 28.2.8 mph

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Physics 221 march 29

The distance between the bumps is about 4.2 m. The natural frequency of the driver and the seat is f = 2 Hz. At what speed does the drive become dangerous?

The drive becomes dangerous when the frequency with which the tractor encounters the bumps equals the natural frequency of the driver and the seat, or when the time interval between hitting the bumps equals the natural period.

f = 2 Hz = 2/s, T = 0.5 s.

v = 4.2 m / T = 8.4 m/s.


Traveling waves

Traveling waves

Periodic, sinusoidal waves in one dimension:

The displacement as a function of position and time is given by

y(x,t) = A sin(kx-ωt + φ)

for a wave traveling in the +x direction, and by

y(x,t) = A sin(kx+ωt + φ)

for a wave traveling in the -x direction.

k is the wavenumber, k = 2π/λ.

ω = 2π/T = 2πf is the angular frequency.

φ is called the phase constant.

λ = wavelength, f = frequency, T = period, v = λ/T = λf = ω/k = speed.

The energy E transported by the wave is proportional to  A2.

The power P is proportional to A2v.

For waves on a string:


Physics 221 march 29

Extra credit:A wave travelling in the +x direction is described by the equationy = 0.1 sin (10 x − 100 t)where x and y are in meters and t is in seconds.Calculate(i) the wavelength, (ii) the period,(iii) the speed, and(iv) the amplitude of the wave

  • (i) 10 m, (ii) 0.100 s, (iii) 1 m/s, (iv) 0.1 m

  • (i) 0.63 m, (ii) 15.9 s, (iii) 10 m/s, (iv) 0.2 m

  • (i) 0.1 m, (ii) 0.01 s, (iii) 10 m/s, (iv) 0.1 m

  • (i) 0.63 m, (ii) 0.063 s, (iii) 10 m/s, (iv) 0.1 m

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Physics 221 march 29

Extra point:If you double the wavelength λ of a wave on a string under fixed tension, what happens to the wave speed v and the wave frequency f?

  • v is doubled and f is doubled.

  • v is doubled and f is unchanged.

  • v is unchanged and f is halved.

  • v is unchanged and f is doubled.

  • v is halved and f is unchanged.

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Physics 221 march 29

A travelling wave is described by the equationy(x,t) = (0.003) cos(20 x + 200 t )where y and x are measured in meters and t in secondsWhat is the direction in which the wave is travelling?

  • The + x direction

  • The –x direction

  • This is a standing wave

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Which of the following wave functions describe a wave that moves in the x direction

Which of the following wave functions describe a wave that moves in the -x-direction?

  • y(x,t) = A sin (–kx – ωt)

  • y(x,t) = A sin (kx + ωt)

  • y(x,t) = A cos (kx + ωt)

  • both 2. and 3.

  • all of them

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Physics 221 march 29

The four strings of a musical instrument are all made of the same material and are under the same tension, but have different thicknesses. Waves travel

  • fastest on the thickest string.

  • fastest on the thinnest string.

  • at the same speed on all strings.

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Standing waves on a string

Standing waves on a string

For a string fixed on both endsa standing waveforms when

an integral number of half wavelengths fit into the length of

the string.


Physics 221 march 29

Extra credit:A particular violin string plays at a frequency of 440 Hz.If the tension is increased by 8.0%, what is the new frequency?

  • 440 Hz

  • 407 Hz

  • 475 Hz

  • 423 Hz

  • 457 Hz

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