If you stand on a bathroom scale, the spring inside the scale ...

Question

If you stand on a bathroom scale, the spring inside the scale compresses 0.60 mm, and it tells you your weight is 760 N. Now if you jump on the scale from a height of 1.0 m, what does the scale read at its peak? Assume Hooke’s law holds.

Accepted Answer

Welcome back. Everyone in this problem. A fruit basket with a total mass of 3 kg is gently placed on a kitchen scale, compressing the scales spring by two millimeters which accurately displays the basket's weight. If the same basket is accidentally dropped from a height of 0.5 m onto the scale. What maximum weight does the scale display at the moment of greatest compression for our answer choices? A says that it's 4.9 multiplied by 10 squared newtons. B 5.8 multiplied by 10 squared newtons. C 6.6 multiplied by 10 squared newtons and D 7.2 multiplied by 10 squared newtons. Now let's try to visualize what's going on here. So basically, we have two scenarios. Let's say that the first scenario. In scenario, a we place our basket gently on the spring. OK, that it compresses it by two millimeters so we can represent that two millimeter distance here. OK. But then in our second scenario, OK, we drop our basket from a height of 50 sorry, a height of 0.5 m. OK? Height of 0.5 m. It's actually, you know what? Let me show it in the air before it drops. So we're gonna drop our basket. OK. We're gonna drop our basket onto the scale from a height of 0.5 m. And when we do, OK, then we want to figure out what maximum weight the scale displays at the greatest moment of compression. In other words, we don't know what our compression is. OK? We don't know what X is. Let me put that in red or blue to show that we don't know. But we do want to figure out what that maximum weight is. In other words, what that force is going to be equal to and let's call that force F max. Now, if we're going to figure out that force, let's ask ourselves, what do we know about force and uh the, the compression or maximum or greatest compression here? What, what do we know that can help us? Well, recall that by hook's law, it tells us that the force applied is directly proportional to the compression, which says that F equals K multiplied by X where K is the spring constant. So that means then for our maximum weight F max is going to be equal to K multiplied by X max. That is the maximum compression. Now, assuming that air resistance is negligible and that energy is conserved during the drop that allows us to equate the potential energy at the beginning of the drop to the energy stored in our spring. So this maximum force exerted by the spring is going to be the same as the maximum weight displayed by the scale. That is why we were able to label this force as F max. But the problem is we don't know the spring constant. OK. And neither do we know our maximum compression. So if we can figure both of those out, then we'll be able to find a maximum force. So let's start by calculating the spring constant. No, because we have two scenarios, we can use that same formula. OK. We can use that same formula to help us figure out the value of K. Recall that in the 1st, 1st scenario since the spring supports the basket in equilibrium, then the force exerted by the spring is equal to the weight of the basket. In other words, of spring is going to be equal to W since that's the case again, based on Hook's law, that means KX because it's still going to have the same spring constant. We're still talking about the same scale is going to be equal to MG. Therefore K the spring constant is going to be equal to MG divided by X so far. OK. We know that the mass of our basket is 3 kg. We'll stick the acceleration due to gravity to be 9.8 m per second squared. And we know that initially it was compressed at a distance of two millimeters which in si units that would be 0.002 m. OK. So all of that is being divided by 0.002 m. OK? And when we do that, we get our spring constant K to be equal to 14,700 newtons per meter. Thus, we have the spring constant now that we have the spring constant. Let's see if we can figure out the maximum compression of the spring X max K. Now let's again, let's ask ourselves as we mentioned earlier that the energy conserved during the drop is equal to the, the sorry, the energy is conserved during the job. And that means the potential energy at the beginning of the job is equal to the energy stored in the spring. In other words, the gravitational potential energy we're talking about our scenario be now is going to be equal to the potential energy stored in the or or spring here or elastic potential energy. What do we know about elastic potential energy? Well, we know that potential energy due to gravity equals mass multiplied by gravity multiplied by height and elastic potential energy equals half kx squared. In this case, we're talking about X max. So now we can use this idea to help us figure out what X max is going to be because that means then that X max squared equals two MGH divided by K, which means that X max, if we square root both sides is going to be the square root of two MGH divided by K. And when we substitute that as we know the value of KMG and H then that means it's going to be equal to the square root of two multiplied by 3 kg, multiplied by 9.8 m per second squared multiplied by the height that it was dropped, which is 0.5 m. And all of that is being divided by the spring constant K, which we just found to be 14,700 newtons per meter. Thus, we can use this expression to find X max the maximum compression. And when we do that, OK, then we get X max to be equal to 0.0447 m. OK. Now this is good because now we've been able to find the spring constant K, the maximum compression X max. And we can substitute these to solve now for F Max because no, that means F max which was equal to K multiplied by X max is now going to be equal to 14,700 newtons per meter multiplied by 0.0447 m. OK? And when we work that out, then we should get it to be equal to 657.09 Newton, which when we add up to two significant figures, it's 6.6 multiplied by 10 squared Newton. Thus, the weight that is going to be shown on the scale is 6.6 multiplied by 10 squared newtons. When we look back at our answer choices, that means our C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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If you stand on a bathroom scale, the spring inside the scale compresses 0.60 mm, and it tells you your weight is 760 N. Now if you jump on the scale from a height of 1.0 m, what does the scale read at its peak? Assume Hooke’s law holds.

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