Explain the link between simple harmonic motion and waves. Get a feel for the force required to maintain this periodic motion. . selection of the mass of the object in the previous problem if you did not wish the new period to be greater than. Oscillations are of central importance to the study of sound and to is to attach a mass to the end of a spring and then set it into motion. It's because both t and the period (T) have units of seconds; this means their ratio has no units. When the mass is at its equilibrium position, the force of gravity (the. The period of a mass m on a spring of spring constant k can be calculated as was given to this relationship between force and displacement is Hooke's law.
We find that both of these points are the same distance from the equilibrium position. This quantity is called the amplitude of the motion. There is a simple correspondence between the terms we've used to describe simple harmonic oscillations and those we use to describe sound. The frequency of oscillations is related to the pitch of sound.
The amplitude of oscillation is related to the loudness of sound. We'll discuss this in more detail later in the semester.
Sound generally involves the superposition of many different pitches, corresponding to describing general oscillations as a superposition of simple oscillations at different frequencies. The motion of a simple harmonic oscillator is related to a pure tone single frequency in sound.
We can quantitatively measure the position of the mass versus time. Graphically, the position versus time looks like When working with the equation describing position versus time, we will end up dealing with trigonometric functions. You might need to review them. Otherwise, you would be specifying an angle for the "sin" or "cos" function in degrees. In the above formula, the "angle" for the "sin" function has no units. It's because both t and the period T have units of seconds; this means their ratio has no units.
An angle specified like this is said to be given in radians even though that's not really a physical unit. You need to make sure you see "R" or "RAD" in your calculator's display. Bring it to my office if you need help setting it up! Oh yeah, there is another units conversion factor to convert an angle in radians to one in degrees.
Mass on a Spring
That is shown in the next figure The maximum velocity vmax is related to the amplitude of the oscillation by We'll see why this is so, when we talk about energy considerations for a simple harmonic oscillator.
We can analyze the velocity versus time graph using techniques we discussed earlier, to deduce the acceleration of the mass at each instant in time. See if you can convince yourself of the following: What causes the motion? The motion of the mass is caused by the force exerted on it by the spring. Before relating that force to the motion of the oscillator, let's understand the force exerted by a spring when there is no motion.
We can do that by hanging different masses onto the the end of a spring mounted vertically, and measuring where the equilibrium position is for each mass.
With careful measurements, we find a linear relationship between these equilibrium positions and the mass attached to the spring. The minus sign simply is telling us the direction of the force. The spring exerts a force on the mass in the opposite direction to the displacement from equilibrium. If we stretch a spring, the force tries to pull the mass back to equilibrium i. If we compress the spring, the force tries to push the mass back to equilibrium.
That minus sign in our mathematical equation for the force is actually the critical thing that leads to simple harmonic oscillation. The support force Fsupport balances the force of gravity.
It is supplied by the air from the air track, causing the glider to levitate about the track's surface. The final force is the spring force Fspring. As discussed above, the spring force varies in magnitude and in direction. Its magnitude can be found using Hooke's law. Its direction is always opposite the direction of stretch and towards the equilibrium position. As the air track glider does the back and forth, the spring force Fspring acts as the restoring force.
Motion of a Mass on a Spring
It acts leftward on the glider when it is positioned to the right of the equilibrium position; and it acts rightward on the glider when it is positioned to the left of the equilibrium position. Let's suppose that the glider is pulled to the right of the equilibrium position and released from rest. The diagram below shows the direction of the spring force at five different positions over the course of the glider's path.
As the glider moves from position A the release point to position B and then to position C, the spring force acts leftward upon the leftward moving glider. As the glider approaches position C, the amount of stretch of the spring decreases and the spring force decreases, consistent with Hooke's Law.
Despite this decrease in the spring force, there is still an acceleration caused by the restoring force for the entire span from position A to position C. At position C, the glider has reached its maximum speed. Once the glider passes to the left of position C, the spring force acts rightward. During this phase of the glider's cycle, the spring is being compressed. The further past position C that the glider moves, the greater the amount of compression and the greater the spring force.
This spring force acts as a restoring force, slowing the glider down as it moves from position C to position D to position E.
By the time the glider has reached position E, it has slowed down to a momentary rest position before changing its direction and heading back towards the equilibrium position. During the glider's motion from position E to position C, the amount that the spring is compressed decreases and the spring force decreases.
There is still an acceleration for the entire distance from position E to position C.
Now the glider begins to move to the right of point C. As it does, the spring force acts leftward upon the rightward moving glider. This restoring force causes the glider to slow down during the entire path from position C to position D to position E. Sinusoidal Nature of the Motion of a Mass on a Spring Previously in this lessonthe variations in the position of a mass on a spring with respect to time were discussed.
At that time, it was shown that the position of a mass on a spring varies with the sine of the time. The discussion pertained to a mass that was vibrating up and down while suspended from the spring.
The discussion would be just as applicable to our glider moving along the air track. If a motion detector were placed at the right end of the air track to collect data for a position vs. Position A is the right-most position on the air track when the glider is closest to the detector.
The labeled positions in the diagram above are the same positions used in the discussion of restoring force above. You might recall from that discussion that positions A and E were positions at which the mass had a zero velocity. Position C was the equilibrium position and was the position of maximum speed.
If the same motion detector that collected position-time data were used to collect velocity-time data, then the plotted data would look like the graph below. Observe that the velocity-time plot for the mass on a spring is also a sinusoidal shaped plot. The only difference between the position-time and the velocity-time plots is that one is shifted one-fourth of a vibrational cycle away from the other.
Also observe in the plots that the absolute value of the velocity is greatest at position C corresponding to the equilibrium position. The velocity of any moving object, whether vibrating or not, is the speed with a direction. The magnitude of the velocity is the speed. The direction is often expressed as a positive or a negative sign. In some instances, the velocity has a negative direction the glider is moving leftward and its velocity is plotted below the time axis. In other cases, the velocity has a positive direction the glider is moving rightward and its velocity is plotted above the time axis.
You will also notice that the velocity is zero whenever the position is at an extreme.