A 6.0-kg box moving at 3.0 m/s on a horizontal, frictionless surf...

Question

A 6.0-kg box moving at 3.0 m/s on a horizontal, frictionless surface runs into a light spring of force constant 75 N/cm. Use the work–energy theorem to find the maximum compression of the spring.

Accepted Answer

Hey everyone. So today we're dealing with the problem about the compression of the spring and the work and kinetic energies of the blocks that are hitting it. So in this question, we're being told That in a competition between kids, a block with a mass of five kg is thrown towards the spring along a flat surface with negligible friction. And the first spring has a force constant K of 70 newtons per centimeter. If one kid throws the block at 2m/s, we're being asked to find the greatest compression observed on the spring. So let's conceptualize this real quick in the competition. This is our flat surface and this is our spring, Very bad spring. But we have a five kg block that has been pushed as being pushed or thrown rather towards spring in order to form a compressed spring. And we're trying to find that given the velocity, the initial velocity of the throne block being two m/s, what will be the change in distance, change in distance here? So before we get to that, we need to take into account that there are two parts of the question. There is the work done by the spring force as well as the work done by the kid by the block he has thrown. So let's start with the block. The network done according to the um work energy theorem, is that the net work will be the final kinetic energy minus the initial kinetic energy. Now the final kinetic energy kinetic energy final would just be zero because that would be a point right here, when the spring is at maximum compression. The final will be zero because for a split second the block will not move. It has no velocity. So that means that the work done here will simply be zero minus the definition of kinetic energy half M. V. Naught squared. Because we're dealing with the initial kinetic energy. The other definitely needs to look out for. Is that the work done by the spring, Work done by the spring is equal to negative 1/ K. The force constant. And recall that K is equal to 70 newtons per centimeter. K multiplied by the distance compressed squared. So since these are both equitable to network, we can equate them both to each other as well. So we get one half. Okay squared Is equal to negative 1/2 M V. Not squared. Rearranging to solve for the X. Term. Because we are asking what is the greatest compression observed on the string? That is what we're looking for rearranging. We get that X squared is equal to M. V. Not squared over. Okay, so X is therefore, so X is therefore the square root of Mv not squared over. A can. Now we can go ahead and start solving but there's one last thing that we need to uh recall. So if K Is equal to 70 newtons per centimeter while we want everything to be in S. I. units primarily the cm is not an S. I unit. We need to convert this two newtons per meter. So 70 newtons per meter or per centimeter. We can recall that for every one m we have or oops, we can recall that for every Yeah, for every one m we have because we want meters to be in the bottom. We have 100 centimeters so 100 centimeters per meter. So centimeters will cancel out 70 times 100. And we get that. The force constant K. Is 7000 newtons per meter. So now plugging this into our equation, we get The mass is five kg, 5 kg multiplied by two m/s squared divided by 7000 Newtons Per meter. And then we take the square root of that entire quantity right? That little better. The square root of that entire quantity. And solving for X, we get a final answer Of 5.35 cm. Excuse me? It's 5.35 cm. Due to the fact that the answer in meters 0. m multiplied by the universe conversion factor. We have 100 centimeters for every one m will give us this answer note that for this equation for this equation, oops, this one my bad. The compression of the spring increases when either the initial velocity or the mass increases and the compression of the spring will decrease when the compression constant the force constant increases because that indicates a stiffer spring. So The greatest compression observed on the string will be answer choice C5.35 cm. I hope this helps, and I look forward to seeing you all in the next one.

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