To verify the formula for the period, T, of an oscillating mass-spring system. Equipment "mass." Mathematically, Fs = - kx, where k is the spring constant. . Calculate a percent difference for the k values obtained by the two methods. Note that. 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. It's because both t and the period (T) have units of seconds; this means their ratio has no units. When the mass is at its equilibrium position, the force of gravity (the. To determine this quantitative relationship between the amount of force and the amount of . The equation that relates kinetic energy (KE) to mass (m) and speed (v) is . As we begin our study of waves in Lesson 2, concepts of frequency.
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.
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.
Period dependence for mass on spring (video) | Khan Academy
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. Note that the equation for acceleration is similar to the equation for displacement.
The acceleration can in fact be written as: All of the equations above, for displacement, velocity, and acceleration as a function of time, apply to any system undergoing simple harmonic motion.
What distinguishes one system from another is what determines the frequency of the motion. We'll look at that for two systems, a mass on a spring, and a pendulum.
The frequency of the motion for a mass on a spring For SHM, the oscillation frequency depends on the restoring force. This is the net force acting, so it equals ma: This gives a relationship between the angular velocity, the spring constant, and the mass: The simple pendulum A simple pendulum is a pendulum with all the mass the same distance from the support point, like a ball on the end of a string. Gravity provides the restoring force a component of the weight of the pendulum.
Summing torques, the restoring torque being the only one, gives: The amplitude, changes in the amplitude do not affect the period at all. So pull this mass back a little bit, just a little bit of an amplitude, it'll oscillate with a certain period, let's say, three seconds, just to make it not abstract. And let's say we pull it back much farther. It should oscillate still with three seconds. So it has farther to travel, but it's gonna be traveling faster and the amplitude does not affect the period for a mass oscillating on a spring.
This is kinda crazy, but it's true and it's important to remember. This amplitude does not affect the period.
Period dependence for mass on spring
In other words, if you were to look at this on a graph, let's say you graphed this, put this thing on a graph, if we increase the amplitude, what would happen to this graph? Well, it would just stretch this way, right? We'd have a bigger amplitude, but you can do that and there would not necessarily be any stretch this way.
If you leave everything else the same and all you do is change the amplitude, the period would remain the same. The period this way would not change. So, changes in amplitude do not affect the period. So, what does affect the period?Simple Harmonic Motion
I'd be like, alright, so the amplitude doesn't affect it, what does affect the period? Well, let me just give you the formula for it. So the formula for the period of a mass on a spring is the period here is gonna be equal to, this is for the period of a mass on a spring, turns out it's equal to two pi times the square root of the mass that's connected to the spring divided by the spring constant.
Motion of a Mass on a Spring
That is the same spring constant that you have in Hooke's law, so it's that spring constant there. It's also the one you see in the energy formula for a spring, same spring constant all the way. 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.
Motion of a Mass on a Spring
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.