You push a 3-kg mass against a spring and release it from rest. Its maximum acceleration is 10m/s 2 when pushed back 0.5m. What is the (a)spring constant and (b) restoring force at this point?

Hey, guys. Let me help you out with this one just in case you were struggling. So we've got a 3 kilogram mass that is sitting on a spring. So let me go ahead and draw that out. So I've got this mass that's attached to the spring here, and we're told that we pushed it back and compressed it, pushed it back 0.5 meters. So relative to this equilibrium position, this thing was pushed back a distance of x equals 0.5 meters. And so what happens is once we push this thing back, we're going to assume that we're going to let it go. The only thing that's acting on this is the spring force, And once that happens, it's got an acceleration in this direction that's equal to 10 meters per second squared. So that's basically all we know. And now we're supposed to ask, we're supposed to find out what the spring constant is and also what that restoring force is or the the spring force. Right? So let's go ahead and check that out. So I have what the mass is, mass is equal to 3 kilograms. I've got that the acceleration is equal to 10, and I'm told that the x displacement is equal or the deformation is equal to 0.5. And I'm supposed to find out what k is. So using all of this stuff, what is k? Well, I've gotten it I've got a formula that relates all of these variables together. Remember that the acceleration is equal to negative k over m times x. Now we're just gonna go ahead and throw away this negative sign here because it's just a direction thing. And so now what I'm supposed to do is find out what k is. But I'm told that when x is equal to 0.5 meters that this acceleration is equal to 10. So I'm just gonna use that equation right here. So if I'm trying to figure out what k is, if I go ahead and rearrange this stuff, the x is going to be divided to the other side. The m is gonna be multiplied to the other side. So they're just gonna cross over. So we've got kim. A k is equal to 10 times m, and that's gonna be divided by x. So I'm gonna go ahead and just substitute my values in for that. I've got 10 times 3, and then my deformation is 0.5. So I'm gonna get a k constant, a spring constant, of 60 newtons per meter. So let's take a look at this next question here. What is the restoring force at this point? So now what they're asking for is the spring force. Well, the spring force, which is the applied force, well, once that goes away, is just going to be equal to k times x. So I'm just gonna plug in those variables. So the restoring force here, the spring force is going to be 60 times 0.5. So that's going to be 30 newtons. Alright, guys. That's it for this one. Let's keep going.

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17. Periodic Motion

Spring Force (Hooke's Law)

Physics

17. Periodic Motion

Spring Force (Hooke's Law)

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Multiple Choice

You push a 3-kg mass against a spring and release it from rest. Its maximum acceleration is 10m/s² when pushed back 0.5m. What is the (a)spring constant and (b) restoring force at this point?

60 N/m; 30 N

120 N/m; 45 N

100 N/m; 35 N

30 N/m; 15 N

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Verified step by step guidance

Start by understanding Hooke's Law, which states that the force exerted by a spring is proportional to its displacement: F = -kx, where F is the force, k is the spring constant, and x is the displacement.

Given the maximum acceleration (a) of the mass is 10 m/s², use Newton's second law, F = ma, to find the force exerted by the spring. Here, m = 3 kg and a = 10 m/s², so F = 3 kg * 10 m/s².

Substitute the force calculated from Newton's second law into Hooke's Law to find the spring constant k. Use the displacement x = 0.5 m: F = kx, so k = F/x.

Calculate the restoring force using the spring constant found in the previous step and the displacement. The restoring force is given by F = kx.

Review the options provided and compare the calculated spring constant and restoring force to determine which pair matches your calculations.

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You push a 3-kg mass against a spring and release it from rest. Its maximum acceleration is 10m/s2 when pushed back 0.5m. What is the (a)spring constant and (b) restoring force at this point?

Watch next

Spring Force (Hooke's Law) practice set

You push a 3-kg mass against a spring and release it from rest. Its maximum acceleration is 10m/s² when pushed back 0.5m. What is the (a)spring constant and (b) restoring force at this point?