Spring launches a mass on an incline plane with friction, find the height

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There's a sign error.

The final potential energy should be less than the initial mechanical energy in the spring by the energy lost due to friction.

If you're subtracting the friction energy from the final energy, it's the same as if you're adding the friction energy to the initial energy. I.e., you're saying that you're expecting the system to gain energy during the process due to the friction. You're counting the friction as free energy, instead of an energy loss.

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Addem
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Addem

Updated on June 15, 2022

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  • Addem
    Addem over 1 year

    So I've calculated the answer to this problem, but my answer is different from my book's, so I'm trying to find the error. A spring with coefficient $k=600N/m$ launches a mass of $1.2kg$ from an initial displacement of $0.15m$. It slides along a frictionless surface and then goes up an inclined plane with coefficient of friction $\mu_{k}=0.2$ and angle $\theta = 30^{o}$. What is the maximum vertical height it accomplishes?

    So I thought, initially the total energy of the system was $\frac{1}{2}kx^{2}$ but when it reaches maximum height it has potential energy $mgh$ and zero kinetic energy, and it has lost the energy equal to the work done by friction, which is $Fd$. The force of friction is $F=\mu_{k}mg\cos\theta$ and the distance $d= \frac{h}{\sin\theta}$ where $h$ is the final vertical height.

    So I get the equation

    $$\frac{1}{2}kx^{2} = mgh - \mu_{k}mg\cos\theta \left(\frac{h}{\sin\theta}\right)\Longrightarrow $$

    $$h = \frac{kx^{2}}{2mg\left(1-\frac{\mu_{k}\cos\theta}{\sin\theta}\right)}$$

    When I plug in all the numbers and compute (done here: Wolfram calculation) I get about .878 which is apparently incorrect by about a factor of 2. Any idea where this went wrong?

  • Addem
    Addem about 9 years
    I'm confused, in the way that I did this, the initial kinetic energy was 0. The initial mechanical energy was due to the spring and there was no other energy--the final mechanical energy was entirely potential energy from gravity, and it was less than the initial mechanical energy because I subtracted the energy lost due to friction. So I'm not seeing the sign error.
  • Addem
    Addem about 9 years
    Oh wait, I think I see what you're saying. I shouldn't be subtracting, I should be adding, because the total energy in the system at the end should be the initial energy 0.5kx^2 minus the energy lost due to friction.