# What type of motion is SHM most like?. S.H.M. and circular motion have a lot in common. Both are periodic. Both have positions described by sine functions.

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What type of motion is SHM most like?

S.H.M. and circular motion have a lot in common. Both are periodic. Both have positions described by sine functions. In circular motion x and y are perpendicular sine functions going simultaneously.

S.H.M. and circular motion have a lot in common. For an oscillator we can write position as a sine function. Δx = A*sin (SOMETHING) Guess what the something is related to???

S.H.M. and circular motion have a lot in common. For an oscillator we can write position as a sine function. Δx = A*sin (SOMETHING) SOMETHING has to vary with time Δx = A*sin (something else * t) something else is called “angular frequency”

S.H.M. and circular motion have a lot in common. Δx = A*sin (“Angular Freq.”* t) Where is Δx = 0?

S.H.M. and circular motion have a lot in common. Δx = A*sin (“Angular Freq.”* t) Where is Δx = 0? At t = 0; t = T/2; t = T Where is sin = 0?

S.H.M. and circular motion have a lot in common. Δx = A*sin (“Angular Freq.”* t) Where is Δx = 0? At t = 0; t = T/2; t = T Where is sin = 0? At 0, ∏, & 2∏

S.H.M. and circular motion have a lot in common. Δx = sin (“Angular Freq.”* t) So, “Angular Freq.”* 0 = 0 “Angular Freq.”* T/2 = ∏ “Angular Freq.”* T = 2∏ Angular Freq. = 2∏/T

What else equals 2∏ divided by the time it takes to go around?

Angular Freq. = 2∏/T What else equals 2∏ divided by the time it takes to go around? Angular frequency is basically the same thing as angular velocity. Both are “ω”.

Angular Freq. = 2∏/T Angular frequency is basically the same thing as angular velocity. Both are “ω”.

Δx(t) So, to put everything in one equation: Δx= A*sin(ω*t) Or Δx= A*sin([2∏/T ]*t) Or Δx= A*sin(2∏* f *t)

Use Sin or Cos waves for this class. But in Physics 2 you will use an offest. Equation: Δx= A*sin(ω*t) Your book sugar coats the equations by making everything start with either x = 0 or +/- A, then you can choose sin or cos. In real life, curves often don’t start at t = 0, 1/4T, 1/2T or 3/4T, so add or subtract and offset amount of t o Δx= A*sin(ω*t - t o )

Industrial example. What is a device that turns circular motion into SHM, or SHM into circular motion?

A crank shaft and piston Engines use these to turn wheels.

Crankshaft Video’s 1 Rotating cam shaft 25 sec Crank Shaft animation on Wiki Put several together 11 sec V-8 Animation showing 4 strokes and 2 cycles Milling a cam shaft 1.5 min Grinding a crankshaft (only show 1 min) Grinding a crankshaft (only show 1 min) Where is the SHM in this video? (Makes for a bumpy ride.)

HW from Last Week. Chart and 2 problems.

Problem A 1 kg block is dropped from a height of 1 m onto a spring with k = 55 N/m. Q1. How far will the spring compress? Q2. What will it’s frequency and period of oscillation be?

In the table, label each +, -, or 0.

Resonance The application of a varying force to an oscillator that matches its frequency.

Resonance We’ve got an oscillator.

Resonance We apply a force every time it reaches the maximum.

Resonance We apply a force every time it reaches the maximum. Resonance Forces replace the energy lost to frictions like air resistance and can increase the energy in a system.

Resonance We apply a force every time it reaches the maximum. Stretches it out a little.

Resonance The next oscillation is bigger.

Resonance The one after that is even bigger.

Resonance And bigger and bigger until the material breaks.

Resonance All objects have “natural frequencies.” They will “wobble” at this frequency if disturbed. A force that resonates at this frequency can be used to break anything.

Anything What made this fall? Resonating wind vibrated it until it collapsed. If the wind was weaker OR STRONGER, the sign would still be standing. Q: Is it the amount of force? No, It’s the timing of the force.

Damped Harmonic Motion In S.H.M, we don’t worry about friction or air resistance, so the amplitude doesn’t change. In the presence of friction, energy dissipates; the amplitude of oscillation decreases as time passes, and the motion is referred to as damped harmonic motion. (Amplitude decreases due to energy loss, often through friction.)

Damped Harmonic Motion Three kinds of damping Underdamping Overdamping Critical Damping

Underdamping -The object still oscillates, but the amplitude is less with each period. (Like the spring on my desk)

Overdamping There is so much friction, there is no oscillation. It just slowly creeps back to equilibrium.

Critical Damping “Just right” amount of friction. Prevents oscillation, but return to equilibrium is fastest. Critical = Boundary between under and over damping. Where would we use this?

Shock Absorber

Are critically damped by design. Prevent oscillation. Return to equilibrium as quickly as possible.

Shock Absorber After the car bounces, do you want it to keep bouncing? NO! Overdamp, then you can feel all the potholes in the road. Underdamp, then the car shakes forever after hitting a pot hole.

What is another important Simple Harmonic oscillator?

How is a pendulum motion sinusoidal?

Picture shows a moving paper with a pendulum ink pen writing on it.

Simple Pendulum The simple pendulum is another example of simple harmonic motion If you look at the free body diagram, the net force is: F = mg sin θ Pointing back to the center

Simple Pendulum F = - mg sin θ This is almost like Hooke’s Law F=-kΔx Because of “the small angle approximation.”

Small angle approximation For small (<10 deg) angles, we can treat the sine function like a straight line.

Period of Simple Pendulum Q: What’s missing? A: Independent of the amplitude What else is missing?

Period of Simple Pendulum Independent of the amplitude NOTE the lack of mass. As we have seen in the past, mass often cancels out of equations when gravity is the driving force, i.e., roller coaster: PE at top and KE at bottom or skid marks from a car wreck?

Period of Simple Pendulum If the battery or spring that powers the pendulum is running down, does the period change? i.e., does a pendulum clock slow down when the batter starts to wear out?

Period of Simple Pendulum If the battery or spring that powers the pendulum is running down, does the period change? i.e., does a pendulum clock slow down when the batter starts to wear out? No, only amplitude changes. How do you adjust the time on it then?

Period of Simple Pendulum If the battery or spring that powers the pendulum is running down, does the period change? i.e., does a pendulum clock slow down when the batter starts to wear out? No, only amplitude changes. How do you adjust the time on it then? Change L. That’s why they put a mass on the end of the pendulum.

Can wind create a resonance force?

Bridge video 1. Show 1 st 4 minutes, then collapse at 7:35 Bridge video 1. Show 1 st 4 minutes, then collapse at 7:35 Bridge Video 2

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