00:01
For this problem, we're given the situation where there is an object hanging from a string, or a spring, and initially the object is held at rest, where the spring is in its rest position, and the object is just being held there, and that position is designated as y -i.
00:21
And then it's allowed to, the object is released, and it's allowed to start oscillating up and down, and the lowest point reaches 10 centimeters below yi.
00:34
And we're being asked to figure out from that what is the frequency of the oscillation.
00:43
And so when we think about this with the object oscillating, if this is the rest point and the lowest point is 10 centimeters, that means our point of equilibrium is actually going to be right at 5 centimeters to half light point.
00:58
Between the highest and the lowest point.
01:02
So we know that our amplitude is going to be five centimeters, or we can just go ahead and put that right into meters, so 0 .05 meters is going to be our amplitude.
01:18
At this point, when our object is at equilibrium, that is going to be when the force of the spring, pulling the object upward is going to be equal to the force of the weight or mass and gravity, pulling it downward.
01:37
So we can write the equation the spring constant times the displacement and this situation is equal to the mass times gravity.
01:54
We're going to be rearranging this to actually solve for our spring constant because our spring constant in this, case is going to equal mass times gravity divided by the position.
02:11
And in this case, our position is actually our amplitude.
02:16
So we're trying to find frequency.
02:17
Well, how does this get us to frequency? well, omega, we know that omega equals 2 pi times our frequency.
02:26
Omega also equals the square root of our spring constant divided by our mass.
02:34
Mass.
02:35
So we can now actually take this and substitute it in for that term and take these two equations and make them equal to each other.
02:46
So in that case, we end up with having the equation 2 pi f equals the square root of mass times gravity divided by displacement divided by mass.
03:12
So in this case our masses are going to end up dividing out.
03:20
So we can actually rewrite this equation as 2 pi f equals the square root of gravity divided by our displacement.
03:33
And then taking that even further, we get frequency, equals the square root of gravity divided by amplitude divided by 2 pi.
03:47
And when we substitute in our known values, we have the square root of 9 .8.
03:54
Again, we're going from the point of equilibrium, so we're going to use our amplitude, so 0 .05.
04:01
All of that divided by 2 pi.
04:04
And we get that our frequency equals 2 .23 hertz.
04:14
The second part of this question is asking us to find the velocity of the object, when the displacement is 8 centimeters below the initial position.
04:28
So below y -i.
04:31
So if we let the zero potential energy be at the initial position of the when the spring is unstretched and we let the downward direction actually be in the positive y axis, then our initial and potential kinetic energies are zero.
04:46
And at position y, the potential and kinetic energies can be determined with the following equations, where potential energy equals 1 half k y squared minus mass times gravity times y times y and kinetic energy is one half mv squared.
05:12
Now, if we take those two and make them equal to zero, 0 equals 1 half ky squared minus mgyy plus 1 half mv squared, we can actually solve for our velocity.
05:33
So rearranging our equation to solve for velocity, we would have velocity equals the square root of 2gy minus k over mass times y squared.
05:54
Now just as before, we can substitute in for k.
05:59
We can change k into mass times gravity over our position...