To determine this quantitative relationship between the amount of force and . Observe that the velocity-time plot for the mass on a spring is also a sinusoidal shaped plot. . As we begin our study of waves in Lesson 2, concepts of frequency. The period of a mass m on a spring of spring constant k can be calculated as .. The next figure shows the basic relationship between uniform circular motion. to measure the spring constant of the springs using Hooke's Law Figure 1. If the stretch is relatively small, the magnitude of the elastic force is where k is a constant, usually called spring constant, and Δx is a stretch (the difference between new (x) and Adding mass to the system would decrease its resonant frequency.
This is the formula for the period of a mass on a spring. Now, I'm not gonna derive this because the derivations typically involve calculus.
If you know some calculus and you want to see how this is derived, check out the videos we've got on simple harmonic motion with calculus, using calculus, and you can see how this equation comes about. But for now, I'm just gonna quote it, and we're gonna sort of just take a tour of this equation. So, the two pi, that's just a constant out front, and then you've got mass here and that should make sense. Why does increasing the mass increase the period? Look it, that's what this says. If we increase the mass, we would increase the period because we'd have a larger numerator over here.
That makes sense 'cause a larger mass means that this thing has more inertia, right. Increase the mass, this mass is gonna be more sluggish to movement, more difficult to whip around.
If it's a small mass, you can whip it around really easily. If it's a large mass, very mass if it's gonna be difficult to change its direction over and over, so it's gonna be harder to move because of that and it's gonna take longer to go through an entire cycle. This spring is gonna find it more difficult to pull this mass and then slow it down and then speed it back up because it's more massive, it's got more inertia. That's why it increases the period.
That's why it takes longer. So increasing the period means it takes longer for this thing to go through a cycle, and that makes sense in terms of the mass. How about this k value? That should make sense too. If we increase the k value, look it, increasing the k would give us more spring force for the same amount of stretch. So, if we increase the k value, this force from the spring is gonna be bigger, so it can pull harder and push harder on this mass.
And so, if you exert a larger force on a mass, you can move it around more quickly, and so, larger force means you can make this mass go through a cycle more quickly and that's why increasing this k gives you a smaller period because if you can whip this mass around more quickly, it takes less time for it to go through a cycle and the period's gonna be less.
That confuses people sometimes, taking more time means it's gonna have a larger period.
Motion of a Mass on a Spring
Sometimes, people think if this mass gets moved around faster, you should have a bigger period, but that's the opposite. If you move this mass around faster, it's gonna take less time to move around, and the period is gonna decrease if you increase that k value.
So this is what the period of a mass on a spring depends on. Note, it does not depend on amplitude. So this is important.
No amplitude up here. Change the amplitude, doesn't matter.
It only depends on the mass and the spring constant. Again, I didn't derive this. If you're curious, watch those videos that do derive it where we use calculus to show this. Something else that's important to note, this equation works even if the mass is hanging vertically. So, if you have this mass hanging from the ceiling, right, something like this, and this mass oscillates vertically up and down, this equation would still give you the period of a mass on a spring.
You'd plug in the mass that you had on the spring here. You'd plug in the spring constant of the spring there. This would still give you the period of the mass on a spring. In other words, it does not depend on the gravitational constant, so little g doesn't show up in here.
Little g would cause this thing to hang downward at a lower equilibrium point, but it does not affect the period of this mass on a spring, which is good news. This formula works for horizontal masses, works for vertical masses, gives you the period in both cases. So, recapping, the period of a mass on a spring does not depend on the amplitude.
You can change the amplitude, but it will not affect how long it takes this mass to go through a whole cycle. And that's true for horizontal masses on a spring and vertical masses on a spring.
The period also does not depend on the gravitational acceleration, so if you took this mass on a spring to Mars or the moon, hung it vertically, let it oscillate, if it's the same mass and the same spring, it would have the same period. It doesn't depend on what the acceleration due to gravity is but the period is affected by the mass on a spring. Sound itself correponds to oscillations of the air pressure. Sound is produced by many different kinds of oscillators that share similar properties to those we will describe today.
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.
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.
Period dependence for mass on spring
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. 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.
Motion of a Mass on a Spring
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.