(II) Consider the track shown in Fig. 8–39. The section AB is ...

Question

(II) Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25 . The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine:

(d) the stiffness constant k for the spring.
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(d) the stiffness constant k for the spring.

Accepted Answer

Hey, everyone in this problem, a toy truck with a mass of 1.8 kg is released from rest from the top of a sand hill. As illustrated below. Upon reaching point Y, the toy truck compresses the spring like device by 0.15 m. And we're asked to calculate the stiffness constant K for the spring like device. The velocity of the truck at point Y is 3.5 m per second and the portion from Y to Z is frictionless. We have four answer choices all in newtons per meter option. A 42 option B 150 option C 280 option D 980. Ok. So we look at our diagram, we have this hill where our truck is on and it's starting at point W and the hill goes down until we get to point X. We continue straight. We get to point Y which is at the start of this spring. Yeah. And then we have this spring device at maximum compression that gets to point Z. Now we're told that this portion from Y to Z is frictionless and what that tells us is that the mechanical energy must be conserved. So the mechanical energy em me at point Y must be equal to the mechanical energy em me at point Z. Now what makes up the mechanical energy we have kinetic energy and potential energy. In this case, a when we're going from point Y to Z, the height of those points is the same. So we don't have to worry about the gravitational potential energy. We can consider that to be height zero. So there's no gravitational potential energy there, but we do have the potential energy due to this spring like device. And so for each term, we're gonna have kinetic energy K plus the potential energy due to the spring us. So Ky plus us, Y is equal to KZ plus us. Z kinetic energy is one half MV squared. Potential energy is one half KX squared when we're talking about the potential energy of a spring. And so what we get is one half mvy squared plus one half K. Xy squared is equal to one half MVZ squared plus one half. OK, XZ squared. Now, at the beginning at point Y we have speed, we're told what the speed is 3.5 m per second. So we know that we have kinetic energy, but we haven't started compressing that spring yet. We're just at the very start of that spring. And so Xy is actually zero, the potential energy due this spring is therefore zero at the end at point Z when we get to the compression of 0.15 m and that's the maximum compression of this spring. And at the maximum compression, the speed is going to be zero. Ok. The truck is gonna momentarily stop as it goes from compressing that spring to the spring kind of flinging it back the other direction. And so momentarily it's gonna come to a stop and the speed VZ is going to be zero. So what we're really doing is we're taking all of that kinetic energy at point Y and we're converting it into spring potential energy at point Z. Now, this constant of one half that we have on both sides that will divide it. We're just left with mvy squared is equal to KXZ squared. We're looking for this cater, OK? We know all the other values. So let's go ahead and substitute them and we get 1.8 kg multiplied by 3.5 m per second. All squared is equal to K multiplied by 0.15 m squared. We can divide both sides by 0.15 m squared. We get that K is equal to 1.8 kg multiplied by 3.5 m per second, all squared divided by 0.15 m squared. And when we work this out, we get this spring constant or this stiffness constant K of 980 newtons per meter. And that is the answer we were looking for comparing this to the answer choices we were given. We can see that this corresponds with option D. Thanks everyone for watching. I hope this video helped you in the next one.

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