Motion of a Mass on a Spring
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. When working with the equation describing position versus time, we It's because both t and the period (T) have units of seconds; this means their ratio has no units. To verify the formula for the period, T, of an oscillating mass-spring system Hooke's law simply states that for a linear spring the spring force, Fs, is proportional . Calculate a percent difference for the k values obtained by the two methods. Calculating the natural frequency of a simple harmonic oscillator Here, k is the spring constant for the spring in question and m is the mass of.
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.
How to Calculate Natural Frequency | Sciencing
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. 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. Once the glider passes the equilibrium position, it begins to slow down as the spring force pulls it backwards against its motion. By the time it has reached position E, the glider has slowed down to a momentary pause before changing directions and accelerating back towards position C.
We turn first to the study of simple harmonic oscillators, so that we can more fully develop a description of an oscillating system, including understanding the motion in detail, the forces that cause that motion and then look at the energy exchanges. Simple Harmonic Oscillation - the motion The easiest way to make a simple harmonic oscillator is to attach a mass to the end of a spring and then set it into motion.
The mass executes repetitive motion, moving back and forth between two points. What can we do to describe its motion in more detail?
Looking at the system before we set it into motion, we see the mass at rest at a position known as its equilibrium position.
Motion of a Mass on a Spring
We'll use that position as the origin of our coordinate system for quantatively describing its motion. If we tap on the mass while it is in its equilibrium position, the oscillations begin. In words, the mass first moves away from equilibrium in one direction we'll call that the positive directionreaches a maximum displacement from equilibrium where it changes its direction of motion instantaneously coming to restspeeds up as it moves back towards the equilibrium position going in the opposite direction compared to when we tapped itslows down as it passes the equilibrium position until it reaches its maximum negative displacement the same distance from the origin as the maximum positive displacement and then heads back to the origin.
What we've described is one cycle of its oscillation. The oscillation cycles repeat. Quantitatively we can measure the time to complete one cycle. This is called the period of the motion generally abbreviated as T. We could also count the number of cycles that occur in each second. That number, in general, will be a fraction: This measure is called the frequency of the motion abbreviated as f. These two measures of the motion are clearly interrelated: The units of f are cycles per second.
Simple Harmonic Motion
In honor of Heinrich Hertz, we use the units of Hertz abbreviated Hz: We can also easily measure the maximum displacement of the mass in both the positive and negative directions. 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.