Simple harmonic motion
We analyse the linear and inverse relations contained in Hooke's law. how the spring constant changes with the length of the spring . The particularly simple relation between the restoring force and displacement in Hooke's law has a. Angular Frequency = sqrt (Spring constant / (Mass) equation can be calculated by clicking on the active word in the relationship above. on a spring is an example of an energy transformation between potential energy and kinetic energy. be the extension of the spring: that is, the difference between the spring's actual length and its unstretched length. is the so-called force constant of the spring.
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. 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.
Period dependence for mass on spring (video) | Khan Academy
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.
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.
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.
It turns out that the velocity is given by: Acceleration in SHM The acceleration also oscillates in simple harmonic motion.
Period dependence for mass on spring
If you consider a mass on a spring, when the displacement is zero the acceleration is also zero, because the spring applies no force. When the displacement is maximum, the acceleration is maximum, because the spring applies maximum force; the force applied by the spring is in the opposite direction as the displacement. A stiff spring would have a high spring constant.
- Motion of a Mass on a Spring
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. 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.
Homework Help: Finding the Amplitude of a spring (Simple Harmonic Motion)
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. 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. 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.