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What is an oscillator?
An oscillator that moves in a way that returns it to the same position, over and over.
Oscillations are also called vibrations.
What are the 2 types of oscillators we learned about?
Pendulum
Mass on a spring
Although these are 2 very simple systems, they help us to model much more complex systems.
What is a cycle?
A cycle is one complete back-and-forth motion of an oscillator.
What 2 quantities do we use to describe oscillatory motion?
Period and frequency.
Period T is defined as the time it takes for the oscillator to complete 1 cycle (or 1 back-and-forth motion). In other words, it is the time to go and come back to the same place.
Period is measured in unit of Time/Cycle. For example: 1 second/cycle or 1 second ("cycle" is not a real unit).
Frequency f is defined as the number of cycles that occur each unit of time. For example: 10 cycles/second, or 10/second ("cycle" is not a real unit).
What is the best way to measure the period of an oscillator?
The best way to measure the period to use a "photogate" detector, such as the one we used in the simulation of a pendulum. When the oscillator passes through its beam of light once, the clock turns on. When the oscillator comes back the 3rd time, the clock turns off. The 2nd pass is only half the period (see figure).
If you have to measure the period with a stopwatch or digital timer, starting and stopping the clock for one period may give a very imprecise measurement. Instead, measure the time to make 10 oscillations (for example), and then divide the total time by 10. This approach minimizes random uncertainties and reaction time errors.
What is a "Hertz"?
Hertz is a unit of frequency.
1 Hertz = 1 cycle/sec
356 Herts = 356 cycles/sec (or vibrations per 1 second)
What is the relationship between period and frequency?
Period and frequency are inversely related.
Period = 1/Frequency
or
T = 1/f
What causes an pendulum to "come back" to the same position over and over?
A restoring force causes an oscillator to return to the same position.
For a pendulum, the restoring force is gravity (see image).
Tension and the component of gravity along the string cancel. The restoring force is the component of gravity that is tangent to the pendulum path.
The the vertical position, there is zero restoring force. (The reason tension is larger than gravity is because the pendulum is still moving in a circular path. Thus, there is a net force toward the center of the circle - where the pendulum is attached.)
What is "equilibrium"?
Equilibrium is the position where an oscillator feels zero restoring force.
An oscillator moves back and forth with the equilibrium position in the middle.
Sketch the restoring force on a pendulum.
The free-body diagram shows that Tension and Gravity are the 2 forces acting on the pendulum.
The component
What is a Simple Harmonic Motion (SHM)?
Simple Harmonic Motion is a special case of oscillation when the restoring force is directly proportional to the displacement of the oscillator from equilibrium.
Force = constant * displacement
For a mass-spring system, the displacement is x, the distance the spring is stretched from equilibrium.
For a pendulum, the displacement is the angle "theta". The restoring force is proportional to the displacement only when theta is small (less than 10 degrees).
"Equilibrium" is the position where the oscillator has zero restoring force.
Why is Simple Harmonic Motion (SHM) important?
As the name implies, it's "simple". But in what way?
For SHM, position, velocity, and acceleration (force) look like sine waves. Although we didn't work with these in this course, sine waves are very easy to analyze.
Not all oscillations are "simple harmonic" - and these are harder to analyze.
Design an experiment to show that the restoring force in a mass-spring system obeys Hooke's Law.
Hooke's Law states that the spring force is directly proportional to the displacement of the mass from equilibrium.
Fs is directly proportional to x (when the graph of Fs vs. x looks like a straight line)
To see if a spring obeys Hooke's Law, you need to measure Fs and x for difference values of x and make a graph of Fs vs. x.
To measure Fs, you need to find the weight of the hanging mass (see the free body diagram). This is because the spring force Fs is equal to the weight of the mass.
To measure x, you can measure the length without any mass (L0) and subtract it from the length with the hanging mass (L), or x=L-L0.
If the graph looks like a straight line, then the spring obeys Hooke's Law and
Fs is proportional x.
Note: Not all springs or oscillators obey Hooke's Law. The ones that do obey this law have Simple Harmonic Motion. If you stretch a spring beyond a certain point, it loses its elasticity and cannot restore itself to equilibrium.
During oscillatory motion of a mass-spring system, where is the restoring force maximum and minimum?
The restoring force is maximum when the spring is compressed or stretched the maximum distance from equilibrium.
The restoring force is minimum (zero) when the mass is at equilibrium.
During oscillatory motion of a mass-spring system, where is the velocity maximum and minimum?
The velocity is minimum (zero) when the spring is compressed or stretched the maximum distance from equilibrium.
The velocity is maximum when the mass moves through the equilibrium point (it overshoots equilibrium).
Sketch the restoring force on a oscillating mass-spring system at
a) maximum compression
b) equilibrium
c) maximum stretch
The restoring force always point toward equilibrium - to restore the situation to equilibrium.
The magnitude of the force is given by Hooke's Law: Fs = kx
What is a spring constant?
A spring constant measured the "stiffness" of an elastic material.
The symbol for a spring constant is "k".
The units are N/m.
F = kx is Hooke's Law, which says that the restoring force is proportional to the displacement.
What does 10 N/m mean? It means that for every additional 1 m of displacement, the restoring force increases by 10 N.
What is a quick way to measure the spring constant of a spring?
Hang a spring on a lab stand.
Measure the length of the spring, Lo.
Hang a mass on the spring.
Measure the stretched length of the spring, L.
Find the amount of stretch from equilibrium x = L-L0.
Use Hooke's Law to find k:
Fs = kx
k = Fs/x
k = Fg/x because Fs = Fg
Note: To truly test that the spring obeys Hooke's Law, you would need to measure Fg vs. x at several values. Then, you would need to plot the data to see of the data is linear.
Where is the kinetic energy of an oscillator maximum/minimum?
Kinetic Energy = 1/2 mass * velocity^2
Wherever the velocity is greatest, the kinetic energy (KE) is greatest. The velocity is maximum at the equilibrium point, and this is where KE is maximum.
The velocity is minimum at the points of maximum displacement from equilibrium. The KE is minimum here too.
If there is no friction, the total energy of an oscillator PE+KE is constant.
Where is the potential energy of an oscillator maximum/minimum?
Potential energy (PE) is largest when the oscillator is storing the most energy.
For a pendulum, the PE is maximum at the maximum height. PE is minimum at the lowest point, which is equilibrium.
For a mass-spring system, the PE is maximum at the point of maximum compression or stretch of the spring. PE is minimum at equilibrium (zero compression/stretch):
U = 1/2 k x^2
Note that stiffer springs (larger k) store more energy for a given amount of stretch.
If there is no friction, the total energy of an oscillator PE+KE is constant.
What is the technical term for the maximum displacement from equilibrium?
Amplitude
The figure shows the Amplitude for a mass-spring system.
For a pendulum, the amplitude is the maximum angle from the vertical position.
If you plot the position of an oscillator vs. time, what does the graph look like?
This position vs. time graph shows the position of a pendulum over a period of time.
What variables affect the period T (or frequency f) of a pendulum?
Length - longer string results in a longer period.
Acceleration due to gravity "g" - greater gravity results in a shorter period.
Note: The period of a pendulum does not depend on the mass of the bob, or the amplitude of the swing.
What would happen if displace a pendulum from equilibrium in zero gravity?
If you are an astronaut orbiting the Earth, you and the spaceship and everything in the spaceship fall around the earth at the same rate. So you and everything else feels "weightless".
If you move a pendulum from equilibrium and release it in these conditions, the pendulum does not move. It requires gravity as a restoring force.
See this video demonstrating the behavior of springs and pendulum in weightless conditions.
"Space Physics: Weightless Springs and Pendulums | NASA ISS Science Video"
https://youtu.be/JghsCipopIc
What would happen if displace a mass-spring system from equilibrium in zero gravity?
If you are an astronaut orbiting the Earth, you and the spaceship and everything in the spaceship fall around the earth at the same rate. So you and everything else feels "weightless".
If you move a mass-spring system from equilibrium and release it in these conditions, the spring will oscillate just like it would on Earth. The restoring force is due to the spring and does not depend on gravity.
See this video demonstrating the behavior of springs and pendulum in weightless conditions.
"Space Physics: Weightless Springs and Pendulums | NASA ISS Science Video"
https://youtu.be/JghsCipopIc
What variables affect the period T (or frequency f) of a mass-spring system?
Mass - greater mass results in a longer period. This is because the greater inertia for a given spring force results in a smaller acceleration of the mass. Thus, the mass takes a longer time to go back and forth.
Spring constant k, which measures the "stiffness" of the spring. Larger k means a stiffer spring. Stiffer springs produce a larger restoring force.
Note: The period of a pendulum does not depend on the amplitude - the maximum stretch or compression of the spring.
What is resonance?
Each oscillator (which really could be any object) has one or more natural frequencies. If you bang an object, it will vibrate at a specific frequency determined by its material and shape.
Resonance happens when you vibrate an object at its natural frequency. Example: Push a swing at the same frequency it swings.
Important: When you drive an object at its natural frequency, its amplitude gets bigger and bigger.
The Tacoma Narrows bridge collapsed because the wind pushed it at the natural frequency. The amplitude of the vibration kept increasing until it tore the bridge apart.
When an opera singer sings a note at the natural frequency of a wine glass, the amplitude of vibrations of the atoms in the glass increases until the glass explodes.
How do you model data that does not look linear when you graph it?
We did two experiments where the data did not look like a line. In this case, you try squaring or square-rooting one of the variables and plot it vs. the other. (You can try raising a variable to different powers also, or taking log of a variable, if the above do not work. But we didn't need to do that.)
Spring-Mass System:
You measured the period T and the mass m for a given spring (constant spring constant k).
When you plotted T vs. m, the data did not fall along a straight line.
However, when you plotted T^2 vs. m, the data looked linear.
You fit a "best fit" line to this graph and were able to write an equation relating T^2 and m:
T^2 = slope m + y-intercept*
Note: "m" here means mass, not slope.
Most of you found the y-intercept to be very small and set it to zero.
We used a similar procedure to determine the relationship between pendulum period T and its length L, for a given g.
