Harmonic oscillator graph between momentum and displacement method
Depending on the friction coefficient, the system can: Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time underdamped oscillator. Decay to the equilibrium position, without oscillations overdamped oscillator.
The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped. If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.
Mechanical examples include pendulums with small angles of displacement , masses connected to springs , and acoustical systems. As such, there are three dominant forces acting upon the glider. These three forces are shown in the free-body diagram at the right. The force of gravity Fgrav is a rather predictable force - both in terms of its magnitude and its direction.
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. 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 lesson , the 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.
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Harmonic oscillator graph between momentum and displacement method high reward investing
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