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I hope you are doing well :)

When I was practicing, I came upon this question:

On a horizontal frictionless surface there is a Spring with a Spring constant is 50 N/m. Initially, the spring is at its relaxed length and a block is stationary at position x = 0. Then an applied force with a constant magnitude of 3.0 N pulls the block in the positive direction of the x-axis, stretching the spring until the block stops. When that stopping point is reached, what is the position of the block?

The question is similar to to this post: Why equating forces give wrong answer?, except that the question in the other post asked for the maximum displacement, and since the system is dynamic, may not be when the block comes to rest. However, the question in this post directly indicates that the block comes to rest.

I thought that since the block was stationary, the applied force would cancel out with the spring force, leading to zero acceleration, so Fapplied + Fspring = 0 ... so doing that, I got -kx + 3 = 0 and since k = 50, -50x + 3 = 0, so x = 3/50 = 0.06 meters. However, the answer key showed that the Work cancels out, meaning Wapplied + Wspring = 0, and got x = 0.12 meters. I would like to know why the Forces don't cancel out, in this scenario, if the block is literally at rest? It would be awesome if someone could help clear my doubt, at your convenience. Thanks, and have a great day!

Qmechanic
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Yashas Ravi
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  • How did calculate $W_{applied} + W_{spring} = 0$? – Gert Jan 15 '21 at 22:11
  • The question is very poorly worded. – David White Jan 16 '21 at 00:57
  • Hi! Because the block is initially stationary (at x = 0), and the question indicates that the block becomes stationary again, this means that, at that point, Wapplied + Wspring = 0. The question asks what the block's position is when the block is stationary for the 2nd time, meaning when Wapplied + Wspring = 0. Thanks! – Yashas Ravi Jan 16 '21 at 17:16

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The equation of motion is $m \ddot x = F - kx$ where $m$ is mass, x is the extension of the spring from its relaxed position and $F$ is the applied force. Starting from rest at $x = 0$, the solution is $x(t) = {{F(1 - cos( {\sqrt{k \over m}})t)} \over k}$. This is oscillatory motion. $x(t)$ is maximum at each t where $ {\sqrt{k \over m}t} = n \pi$; that is, for every $t = {\sqrt{m \over k}n \pi}$ where n is an odd integer 1, 3,5 ... At each of these times $x = {{2F \over k}} = {6 \over {50}} = 0.12$. The maximum extension is not when the applied force and the spring restoring force are equal, because the mass is moving and has inertia. Notice how much simpler the solution is using energy instead of the equation of motion; however, both approaches yield the same results.

John Darby
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  • Hi, thanks so much for your answer!! I have a small doubt, though. The question asked what the block's position was when the block is at rest, so would that necessarily mean that x(t) is at maximum? Because since the block is at rest, wouldn't the acceleration of the block be zero, meaning that the Spring Force cancels with the applied force? Thanks so much! – Yashas Ravi Jan 16 '21 at 17:13
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    Starting from rest F > kx until x where F = kx; during this time F - kx is accelerating m so when F = kx m has velocity still stretching spring. Then kx exceeds F and de-celerates m until velocity is zero which occurs at maximum x. – John Darby Jan 16 '21 at 20:43
  • Hi, that makes sense, thanks so much! So for any general case where Fnet = 0, is there a way to tell whether an object is stationary or has a nonzero velocity only based on knowing the magnitude of the Forces acting on it? Thanks! – Yashas Ravi Jan 17 '21 at 21:17
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    If Fnet = 0 the object has constant velocity which can be zero velocity. You need more information to know the velocity. – John Darby Jan 17 '21 at 22:07
  • Thanks so much! This was very helpful :) – Yashas Ravi Jan 18 '21 at 23:18