Motion of a Mass on a Spring
The equation for the period (T) of a swinging pendulum is T A stiffer spring with a constant mass decreases the period of oscillation. We have already seen that the stretch of a spring is proportional to the force exerted on a spring. These two ideas are contained in the equation. T = 2(m/k). which gives the period T for a mass m attached to a spring with spring constant k. to measure the spring constant of the springs using Hooke's Law to verify that the period of the SHM is proportional to the square root of the mass and independent of the The minus sign in front of the spring constant in equation 1.
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.
Motion 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.
Period dependence for mass on spring (video) | Khan Academy
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. 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.
Period dependence for mass on spring
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.
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.
As the glider approaches position C, the amount of stretch of the spring decreases and the spring force decreases, consistent with Hooke's Law. Despite this decrease in the spring force, there is still an acceleration caused by the restoring force for the entire span from position A to position C. At position C, the glider has reached its maximum speed.
Once the glider passes to the left of position C, the spring force acts rightward. During this phase of the glider's cycle, the spring is being compressed. 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.
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.
And the speed is least when the displacement of the mass relative to its equilibrium position is the greatest. Energy Analysis of a Mass on a Spring On the previous pagean energy analysis for the vibration of a pendulum was discussed.
Here we will conduct a similar analysis for the motion of a mass on a spring. In our discussion, we will refer to the motion of the frictionless glider on the air track that was introduced above.