Motion of a Mass on a Spring
"Running too fast" means, of course, that the period of the pendulum's swing is too short. You want to increase the period of oscillation (also known as decreasing the frequency), so you Explain how the period of a mass-spring system can be independent of amplitude, This is one of my favorite relationships in physics. The period of a mass m on a spring of spring constant k can be calculated as T=2 π√mk .. Describe relationship between the simple harmonic motion and uniform . A simple pendulum acts like a harmonic oscillator with a period dependent. The connection between uniform circular motion and SHM second, having units of hertz (Hz); the period is how long it takes to make one oscillation. We'll look at that for two systems, a mass on a spring, and a pendulum.
Translated from Latin, this means "As the extension, so the force. If we had completed this study about years ago and if we knew some Latinwe would be famous! The spring constant is a positive constant whose value is dependent upon the spring which is being studied.
A stiff spring would have a high spring constant. This is to say that it would take a relatively large amount of force to cause a little displacement. The negative sign in the above equation is an indication that the direction that the spring stretches is opposite the direction of the force which the spring exerts. For instance, when the spring was stretched below its relaxed position, x is downward.
The spring responds to this stretching by exerting an upward force. The x and the F are in opposite directions. A final comment regarding this equation is that it works for a spring which is stretched vertically and for a spring is stretched horizontally such as the one to be discussed below.
Force Analysis of a Mass on a Spring Earlier in this lesson we learned that an object that is vibrating is acted upon by a restoring force.
Motion of a Mass on a Spring
The restoring force causes the vibrating object to slow down as it moves away from the equilibrium position and to speed up as it approaches the equilibrium position. It is this restoring force which is responsible for the vibration. So what is the restoring force for a mass on a spring? We will begin our discussion of this question by considering the system in the diagram below.
The diagram shows an air track and a glider. The glider is attached by a spring to a vertical support. There is a negligible amount of friction between the glider and the air track. As such, there are three dominant forces acting upon the glider.
These three forces are shown in the free-body diagram at the right. The force of gravity Fgrav is a rather predictable force - both in terms of its magnitude and its direction. The support force Fsupport balances the force of gravity. It is supplied by the air from the air track, causing the glider to levitate about the track's surface.Simple Harmonic Motion, Mass Spring System - Amplitude, Frequency, Velocity - Physics Problems
The final force is the spring force Fspring. As discussed above, the spring force varies in magnitude and in direction. Its magnitude can be found using Hooke's law. Its direction is always opposite the direction of stretch and towards the equilibrium position.
As the air track glider does the back and forth, the spring force Fspring acts as the restoring force. 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. Consider an object experiencing uniform circular motion, such as a mass sitting on the edge of a rotating turntable. This is two-dimensional motion, and the x and y position of the object at any time can be found by applying the equations: The motion is uniform circular motion, meaning that the angular velocity is constant, and the angular displacement is related to the angular velocity by the equation: How does this relate to simple harmonic motion?
An object experiencing simple harmonic motion is traveling in one dimension, and its one-dimensional motion is given by an equation of the form The amplitude is simply the maximum displacement of the object from the equilibrium position. So, in other words, the same equation applies to the position of an object experiencing simple harmonic motion and one dimension of the position of an object experiencing uniform circular motion.
- Investigating a mass-on-spring oscillator
- Period dependence for mass on spring
Note that the in the SHM displacement equation is known as the angular frequency. It is related to the frequency f of the motion, and inversely related to the period T: The frequency is how many oscillations there are per second, having units of hertz Hz ; the period is how long it takes to make one oscillation. Velocity in SHM In simple harmonic motion, the velocity constantly changes, oscillating just as the displacement does.
When the displacement is maximum, however, the velocity is zero; when the displacement is zero, the velocity is maximum. 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.
The Factors That Might Affect the Period of Oscillation | Sciencing
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. Sometimes, people think if this mass gets moved around faster, you should have a bigger period, but that's the opposite.
Chapter 10 Concepts
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.