The relationship between the energy and amplitude of a wave? Derivation? - Physics Stack Exchange
The connection between uniform circular motion and SHM. It might seem The amplitude is simply the maximum displacement of the object from the equilibrium position. The frequency of the motion for a mass on a spring. The total energy of the system depends on the amplitude A: The frequency depends only on the force constant of the spring and the mass: Does this relationship between energy and frequency of oscillation remind you of anything else?. A block of mass m tied to a spring with spring constant k is a) Find Amplitude Α Note: At x = Α the kinetic energy is 0. kA2 = Frequency f = v / λ = Hz.
The answer is two-fold: It's very common, in all different sorts of situations It's easy to describe theoretically Here are just a few examples of SHM: SHM in terms of force, or in terms of potential We can describe the conditions under which SHM will occur in several ways.
In terms of forces, SHM occurs when an object experiences a linear restoring force when displaced from its rest position. In other words, a force which pushes it back towards the rest position opposite to the displacement grows larger as the displacement grows larger, in a linear fashion: Mathematically speaking, We can also look at things graphically. If we make a plot showing position along the horizontal axis and potential energy along the vertical axis, we see a familiar form: As the mass moves, it exchanges kinetic energy for spring potential energy, but the sum of the two remains fixed: The total energy of the system depends on the amplitude A: Note that we can give the system any energy we wish, simply by picking the appropriate amplitude.
The frequency of oscillation, on the other hand, does NOT depend on the amplitude of oscillation; that's why we use pendula to drive clocks, of course. The frequency depends only on the force constant of the spring and the mass: Suppose that we were to make a movie an oscillating spring over many cycles. At which position s would the mass appear in the most frames of the movie? In other words, which is the most probable position to find a mass attached to a moving spring?
Where is the mass most probably to be seen? The velocity of the mass is largest as it moves through the equilibrium position, and slowest near the outer reaches of its motion the turning points: So we are most likely to find the mass at the limits of its motion, and least likely to find it near equilibrium. This doesn't depend on the amplitude of the oscillation, so the answer is the same for any energy. Starting with the familiar 1-D time-independent version of the equation, we insert for the potential energy U the appropriate form for a simple harmonic oscillator: A good way to start is to move the second derivative over the to left-hand side of the equation, all by itself, and put all other terms and coefficients on the right-hand side.
Re-arrange the equation so that it looks like this: You should end up with this: We need to find a function which will solve this differential equation.
When in doubt, guess. Which of the following wave functions will solve this particular differential equation? 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.
Simple harmonic motion
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.
- Oscillation amplitude and period
- The Simple Harmonic Oscillator
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.Simple Harmonic Motion - Finding the Amplitude and Period of Oscillation
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.
Motion of a Mass on a Spring
You will also notice that the velocity is zero whenever the position is at an extreme. This occurs at positions A and E when the glider is beginning to change direction. So just as in the case of pendulum motionthe speed is greatest when the displacement of the mass relative to its equilibrium position is the least.
And the speed is least when the displacement of the mass relative to its equilibrium position is the greatest. Energy Analysis of a Mass on a Spring On the previous pagean energy analysis for the vibration of a pendulum was discussed. Here we will conduct a similar analysis for the motion of a mass on a spring. In our discussion, we will refer to the motion of the frictionless glider on the air track that was introduced above. The glider will be pulled to the right of its equilibrium position and be released from rest position A.
As mentioned, the glider then accelerates towards position C the equilibrium position.