A spring of equilibrium length L₁ and spring constant k₁ hangs fr...

Question

A spring of equilibrium length L₁ and spring constant k₁ hangs from the ceiling. Mass m₁ is suspended from its lower end. Then a second spring, with equilibrium length L₂ and spring constant k₂, is hung from the bottom of m₁. Mass m₂ is suspended from this second spring. How far is m₂ below the ceiling?

Accepted Answer

Hi everyone in this particular practice problem, we were asked to actually determine the expression for the system's total length where we, we will have a science teacher using two ideal springs and a small sphere. The sphere is going to be having the mass of MS attached to the three ends of the two springs. Then the system is going to be suspended vertically. The upper spring will have an equilibrium length of L U and a spring constant of K U. The lower spring will have an equilibrium length of L L and a spring constant of K L. And the teacher actually added an additional small block of mass M B to the free end of the lower spring. And we're being asked to find, we were asked to find the final expression of the system's total length after the edition of the small block and B. So first, um I'm going to start with creating a diagram of what our system actually looks like. So for this to work, we will model the sphere and the block to be as particles. So the system will then just be our first upper spring and then we will have our first particle, which is the sphere right here and it will have the second lower spring and we'll have our second particle which is going to be our small block just like. So, so the length of both the sphere and the block is negligible because we're modeling that as sphere um modeling those two, both the sphere and the block as particles. So because we are also assuming that both springs are ideal, then we will know that Hooks Law is going to be applicable to this particular um case. In general using Hooks Law for a suspended particle spring, we can use the equation of delta L equals to M G over K where the delta L is going to be the elongation M is the mass suspended or the mass on the end of the spring, G is just the gravitational acceleration and K is going to be the spring constant. So we can use this when the mass is at rest or the system is at rest, which we're uh seeing in this case. So we are asked to calculate the system's total length and the system's total length is given by the total length of our uh diagram here. So the system's total length will correspond to uh the equilibrium length of the upper spring which is going to be L U and then add the, that with the equilibrium length off the lower spring, which is L L plus because there are two different masses. Here, we have to take the account of the elongation of both springs. So to find the total length, we will have to add the elongation of first, the upper uh spring due to mass MS and M B. And also we have to ask the elo uh added, we have to add the elongation of the lower spring due to mass and B only. So because of this, we can determine that the system's total length is actually going to equal to L U plus L L plus delta L U plus delta L L. And we can actually substitute delta L U and delta L L with the equation give uh we found or we recall previously. So total length is going to then equals to the equilibrium length of the upper spring plus the equilibrium length of the lower spring plus the elongation of the upper spring due to mass MS and M B. So this will be MS plus M B multiplied by G over the spring constant for the upper spring, which is K U plus the delta L of the elongation for the lower spring, which is M uh because of mass M B, which is M B multiplied by G over the spring constant of the lower spring, which is K L. So that will be the expression for the total length of our system which will correspond to option C. So option C is going to be the answer to our particular practice problem and that'll be all for this video. If you guys still have any sort of confusion on this one, please make sure to check out our other lesson videos on similar topics and that'll be all. Thank you.

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