A spring is hanging from the ceiling. Attaching a 500 g physics book to the spring causes it to stretch 20 cm in order to come to equilibrium. b. From equilibrium, the book is pulled down 10 cm and released. What is the period of oscillation?

Video transcript

Hey, everyone in this problem, a 250 g cube is attached to a massless ideal spring suspended vertically from a horizontal table at the equilibrium point. The spring is extended by five centimeters with respect to its natural length. The spring is released after being stretched downward by two centimeters from its equilibrium point. And we're asked to calculate the period of oscillation of the cube. We're given four answer choices. Option, a 0.45 seconds. Option B 1.1 2nd, option C 5.6 seconds and option D seconds. Now recall that the period of oscillation T is equal to two pi multiplied by the square root of M divided by K. So in order to calculate the period, the only two things we need to know are the mass in the spring constant K. Now, we're given the mass of this cube 250 g, but we aren't given the spring constant K. So let me write this down, the mass is equal to 250 g. Let's convert to our standard unit of kilograms. So we're gonna multiply that by one kg, OK, per 1000 g, we have 1000 g in every kilogram, the unit of Graham, we divide it. What we're essentially doing is dividing by 1000. So 250 divided by 1000 gives us 0. kg. Mhm. OK. So we've got our mask, we've got it converted into our standard unit, but we need to find our spring constant kit. Now we call that the spring constant K is related to the force of the spring. OK. That's spring force. So let's draw a free body diagram and think about the forces acting. So we have our cube, it's um suspended vertically. So we're gonna have the sprint force acting upwards and we're gonna have the force of gravity acting downwards. OK? We're gonna take up to be our positive direction. Now, let's think about the equilibrium position. OK? At the equilibrium position, the spring is extended by five centimeters. Now, we know at equilibrium, the sum of the forces is going to be equal to zero. Now, what forces do we have acting? Well, we have the spring force acting in the positive direction and the force of gravity acting in the negative direction. So we have F S minus F G is equal to zero, which tells us that these two forces are equal to each other. The spring force F S is equal to the force of gravity. F G. We call it the spring forces given by K multiplied by X, OK. The spring constant multiplied by the stretcher compression and that's equal to the force of gravity, which is equal to M multiplied by G. Now, there's two ways we can go about this, the value that we need is K. OK. That's the variable that we don't know. So we can isolate for K and solve for K. The other thing we could do is when we substitute into our period equation, we actually calculate M divided by K. So instead of finding K, we could calculate M divided by K here and use that value as well. OK. So either way you want to do it, in this case, I'm just going to calculate K on its own and use it. But you could do it either way. And the other way is actually really handy if you aren't given that mass, even if you aren't given that mass, you can calculate M divided by K given G and X and then you can still solve this problem. I'm gonna scroll down and give ourselves some more room to work. So isolating for K, we get K is equal to M G divided by X. The mass M we found was equal to 0.25 kg. The acceleration due to gravity G is 9.8 m per second squared. And this is all divided by the stretch or compression of the string or the spring. Sorry. Now we're told that it extends five centimeters with respect to its natural length at equilibrium. And so that stretch is gonna be five centimeters and X is equal to five centimeters, converting this to the standard unit of meters. We know that in every meter there are 100 centimeters. So we multiply by one m divided by 100 centimeters. The unit of centimeter divides out, we're essentially dividing by 100 and we get 0.5 m. All right. So back to our equation for K, we have K is equal to 0.25 kg multiplied by 9. m per second squared, divided by 0.5 m. And this gives us a spring constant K of newtons per meter. All right. So we have our spring constant K that we needed to find in order to calculate the period T. So now we're gonna get back to that period team and recall again that period is equal to two pi multiplied by the square root of M divided by K. This is equal to two pi multiplied by the square root of 0.25 kg, divided by 49 newtons per meter, which gives us a period to equal to 0. seconds. And that's what we were looking for. Now, in terms of the units, we had kilogram divided by Newton per meter. Recall that a Newton is a kilogram meter per second squared. So in the denominator, we end up with kilogram per second squared. We have kilogram and the numerator. So the kilograms cancel, we're left with per second squared, which is equal to second squared when it's in the denominator. And then we take the square root and we get seconds, which is exactly what we wanted. So going back up to our answer choices and comparing what we found rounding this to two significant digits. We see that the correct answer is option a 0. seconds. Thanks everyone for watching. I hope this video helped see you in the next one.