# Relationship between frequency mass and spring constant lab

displacement of the mass is (+) and the spring force is to the left A very important property of simple harmonic motion is that the period T does not depend on. The object of this virtual lab is to determine the spring constant k. Each of the blue weights has a mass of 50 grams. 50 grams mass is 2 cm displacement. Using this relationship weights are computed for the masses in the table above. Mass on a spring - Where a mass m attached to a spring with spring constant k, will The period of a simple harmonic oscillator is also independent of its amplitude. In this experiment one of the major sources of error is down to the human accuracy, legality or content of the external site or for that of subsequent links.

How accurate can our measurements be? The experiments described here demonstrate the use of a mix of analogue and digital apparatus to measure quantities including mass, length and time. In this experiment one of the major sources of error is down to the human reaction time when measuring the period.

To improve the accuracy on the period, the timings can be taken over multiple oscillations and by averaging over several measurements of the period. To obtain more accurate measurements of the spring constant and the gravitational acceleration, repeated measurements should be taken using various pendulum lengths and masses. Also, measuring period over a longer time frame and hence over multiple oscillations will increase the accuracy since the human error will be a smaller fraction of the recorded time.

It can also be useful to use a pin or tag to act as a fiduciary marker showing the equilibrium position. Assuming simple harmonic motion, the periodic nature of these systems mean that there should be no excuse when it comes to taking multiple measurements! In this episode we look at generating and measuring waves and the use of appropriate digital instruments.

What do your measurements mean? The vibrations and oscillations that surround us in our everyday lives are generally much more complicated than those we encounter in simple harmonic motion.

This means that effects such as damping, which acts to reduce the amplitude by removing energy from the system, are a good example of where simple harmonic motion contributes to improving our day-to-day lives. Whilst simple harmonic motion is a simplification, it is still a very good approximation.

Simple harmonic motion is important in research to model oscillations for example in wind turbines and vibrations in car suspensions. 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.

## Why is simple harmonic motion so important?

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.

### Physics - simple harmonic motion - University of Birmingham

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.

• Determine the Spring Constant

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.

### Motion of a Mass on a Spring

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.

## Motion of a Mass on a Spring

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.