Simple Harmonic Motion

14.2 Simple Harmonic motion (SHM)14-3 Energy in the Simple Harmonic Oscillator14-4 Simple Harmonic Motion Related to Uniform Circular Motion14-5 The Simple Pendulum14-6 The Physical Pendulum and the Torsional Pendulum

Final ExamSection 001:9:30 am class: Dec. 16 at 11:30amSection 002:11:00 am class: Dec. 16 at 8:00am

Questio

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• A mass on a spring is oscillating back and forth on a frictionless surface. Where in the motion is the velocity the largest

• A) at x = +/-A• B) somewhere between x=0 and x= +/-A• C) at x=0• D) we need more info

x=AX=0x=-A

The velocity and acceleration for simple harmonic motion can be found by differentiating the displacement:

14-2 Simple Harmonic Motion

Example 14-4: Loudspeaker.

The cone of a loudspeaker oscillates in SHM at a frequency of 262 Hz (“middle C”). The amplitude at the center of the cone is A = 1.5 x 10-4 m, and at t = 0, x = A. (a) What equation describes the motion of the center of the cone? (b) What are the velocity and acceleration as a function of time? (c) What is the position of the cone at t = 1.00 ms (= 1.00 x 10-3 s)?

14-2 Simple Harmonic Motion

We already know that the potential energy of a spring is given by:

The total mechanical energy is then:

The total mechanical energy will be conserved, as we are assuming the system is frictionless.

14-3 Energy in the Simple Harmonic Oscillator

Questio

n

• The total energy of a frictionless mass-spring oscillator• A) is constant• B) depends on the amplitude of the oscillation• C) both (A) and (B)• D) neither (A) or (B)

If the mass is at the limits of its motion, the energy is all potential.

If the mass is at the equilibrium point, the energy is all kinetic.We know what the potential energy is at the turning points:

14-3 Energy in the Simple Harmonic

Oscillator

The total energy is, therefore,

And we can write:

This can be solved for the velocity as a function of position:

where

14-3 Energy in the Simple Harmonic Oscillator

This graph shows the potential energy function of a spring. The total energy is constant.

14-3 Energy in the Simple Harmonic Oscillator

Example 14-7: Energy calculations.

For the simple harmonic oscillation of mass m=0.3kg (where k = 19.6 N/m, A = 0.100 m, x = -(0.100 m) cos 8.08t, and v = (0.808 m/s) sin 8.08t), determine (a) the total energy, (b) the kinetic and potential energies as a function of time, (c) the velocity when the mass is 0.050 m from equilibrium, (d) the kinetic and potential energies at half amplitude (x = ± A/2).

14-3 Energy in the Simple Harmonic Oscillator

If we look at the projection onto the x axis of an object moving in a circle of radius A at a constant speed υM , we find that the x component of its velocity varies as:

This is identical to SHM.

14-4 Simple Harmonic Motion Related to Uniform Circular Motion

A simple pendulum consists of a mass at the end of a lightweight cord. We assume that the cord does not stretch, and that its mass is negligible.

14-5 The Simple Pendulum

In order to be in SHM, the restoring force must be proportional to the negative of the displacement. Here we have:which is proportional to sin θ and not to θ itself.

However, if the angle is small: sin θ ≈ θ.

14-5 The Simple Pendulum

Therefore, for small angles, we have:

where

The angular frequency, period and frequency are:

14-5 The Simple Pendulum

So, as long as the cord can be considered massless and the amplitude is small, the period does not depend on the mass.

14-5 The Simple Pendulum

14-5 The Simple Pendulum

Example 14-9: Measuring g.

A geologist uses a simple pendulum that has a length of 37.10 cm and a frequency of 0.8190 Hz at a particular location on the Earth. What is the acceleration of gravity at this location?

A physical pendulum is any real extended object that oscillates back and forth.

The torque about point O is:

Substituting into Newton’s second law gives:

14-6 The Physical Pendulum and the Torsional Pendulum

14-6 The Physical Pendulum and the Torsional Pendulum

For small angles, this becomes:

which is the equation for SHM, with

14-6 The Physical Pendulum and the Torsional Pendulum

A torsional pendulum is one that twists rather than swings. The motion is SHM as long as the wire obeys Hooke’s law, with

(K is a constant that depends on the wire.)