A spring is hung from the ceiling. When a block is attached to it...

Question

A spring is hung from the ceiling. When a block is attached to its end, it stretches 2.0 cm before reaching its new equilibrium length. The block is then pulled down slightly and released. What is the frequency of oscillation?

Accepted Answer

Hey, everyone in this problem, a baby hanging toy is made of an ideal spring and a singing bird toy of mass 200 g. The sprint attached to a horizontal ward is suspended vertically. When the B bird toy is attached, the spring is elongated by four centimeters from its rest length. The baby brings a bird toy down, releases it and watches it oscillate. We're asked to calculate the frequency of the toy bird's motion. We're given four answer choices, option A 1.1 Hertz, option B 2.5 Hertz, option C 16 Hertz and option D Hertz. Now we're asked for frequency of the motion. We're called that we can write the period T equal to two pi multiplied by the square root of M divided by K. Now this is the period we're looking for the frequency, but we know that the period is equal to one divided by the frequency. So using this equation, we can write the frequency is going to be equal to one divided by two pi multiplied by the square root of K divided by M OK. We're essentially taking one divided by this period. All right, So, in order to find the frequency, we need the spring constant K and the mass F, now we have the mass M, we don't have a spring costed K. So we need to do something first to find that. Now, recall that the spring constant is related to the force, the spring force. So let's wrap body diagram and think about the forces acting in our system. So we have this toy bird and acting upwards is gonna be this spring force F S acting downwards is gonna be the force of gravity. OK? Because this spring is hanging vertically so that spring force is acting upwards. Gravity, pulling this toy down, we're gonna take up to be the positive direction. Now we're told some information about the rest. OK? When the bird bird toy is attached, the spring is elongated by four centimeters from its rest lengths. So that's just sitting in equilibrium. So what that tells us is that the sum of the forces is equal to zero. OK? It's not moving at that point. And that's before the baby brings it down and has it o Now the sum of the forces will, we have the spring force F S acting in the positive direction, the force of gravity acting in the negative direction. So we get F S minus F G is equal to zero. It tells us that those two forces are equal, the spring force is equal to the force of gravity. No recall that the spring force is equal to K multiplied by X K and the spring constant multiplied by the structure compression. The force of gravity is equal to the mass M multiplied by the acceleration due to gravity and G. So we wanna find the spring constant K here, this is gonna be equal to F G divided by X. Substituting in our values, we have a mass of 200 g. OK? We wanna write this in our standard unit of kilograms. So what we're gonna do is we're gonna multiply and for every one kg, we have 1000 g. So we multiply by one kg divided by 1000 g, the unit of gram divides them. And what we're essentially doing is dividing by 1000 to go from grams to kilograms. Forget that this is equal to 0.2 kg. Now we're gonna do the same for X and we're told that this toy, this spring is elongated four centimeters from its wrestling. And so X is gonna be equal to four centimeters. OK? And again, for every one m, there are 100 centimeters converting this to our standard unit, the unit of centimeter divides up. And what we're doing is dividing by 100 to go from centimeters to meters. But yet four centimeters is equal to 0.4 m. So going back to our calculation for K, we have the mass 0.2 kg multiplied by the acceleration due to gravity 9.8 m per second squared divided by this stretch of 0.4 m. And this gives us a K value of 49 newtons per meter. A kilogram meter per second is a Newton. And then we're dividing by meter. So we get Newton per meter. All right. So we found our spring constant K. Now we can go back to our equation for the frequency and I'm gonna rewrite that just to remind ourselves. So the frequency that we wrote is going to be equal to one divided by two pi multiplied by the square root of K divided by M substituting in our values, the frequency is equal to one divided by two pi multiplied by the square root of 49 newtons per meter divided by 0. kg, which gives us a frequency of 2. Hertz. OK. When we're looking at units, we have a Newton which is kilogram meter per second squared divided by meter divided by kilogram. So inside this square root, what we're left with is per second squared. When we take the square root, we get per second which is equivalent to Hertz. And so we found that our frequency is equal to 2.49 Hertz. I wanna make one quick comment before we wrap this question up. When we were calculating the spring constant K here. Notice that we needed K in order to use it in this equation where we have K divided by M. OK. So instead of calculating K, like we did substituting that in and substituting M in, we could have also calculated K divided by M. From this equation where we have K is equal to M G divided by X. We could have written K divided by M is equal to G divided by X. And used that value in our equation. OK. You'll get the exact same answer. But I just wanted to make a note that sometimes you can calculate K divided by M instead of just K. And that's really handy if you aren't given the value of M as well. And you could, you could still solve this question that way. All right. So we found our frequency 2.49 Hertz. If we go back up to our answer choices, we round to two significant digits, we see that this corresponds with answer choice B 2.5 Hertz. Thanks everyone for watching. I hope this video helped see you in the next one.

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A spring is hung from the ceiling. When a block is attached to its end, it stretches 2.0 cm before reaching its new equilibrium length. The block is then pulled down slightly and released. What is the frequency of oscillation?

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