Motion of a Mass on a Spring
The frequency of simple harmonic motion like a mass on a spring is in the equation can be calculated by clicking on the active word in the relationship above. He also explains what does not affect the period of a mass on a spring (i.e. amplitude whats the difference between convex and concave lens, and how do we. The connection between uniform circular motion and SHM angular velocity is constant, and the angular displacement is related to the angular velocity by the equation: The frequency of the motion for a mass on a spring.
The further past position C that the glider moves, the greater the amount of compression and the greater the spring force. 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. If a motion detector were placed at the right end of the air track to collect data for a position vs.Period dependence for mass on spring - Physics - Khan Academy
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. 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. 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.
This is kinda crazy but something you need to remember. 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.
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.
Simple harmonic motion
So, what does affect the period? 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.
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.
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?
Motion of a Mass on a Spring
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. 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. If you consider a mass on a spring, when the displacement is zero the acceleration is also zero, because the spring applies no force.
- The Swinging Pendulum
- Simple harmonic motion
- Simple harmonic motion in spring-mass systems
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. The acceleration is given by: Note that the equation for acceleration is similar to the equation for displacement.