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May 5th, 2015, 12:13 PM   #1
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Spring question-Conservation of energy

A particle of mass 2 kg is attached to one end of a model spring that ishanging vertically from a fixed point O. The spring has stiffness 4 N m−1and natural length 1 m. The system is oscillating in a vertical line with theparticle below O. In this question use the approximation that themagnitude of the acceleration due to gravity is 10ms−2. Take the point Oas the datum for potential energy, and measure x downwards.
(a) Find an expression for the total mechanical energy function for the system.
(b) When the particle is 2 m below A, it has speed 1 m s−1. Use conservation of mechanical energy to establish whether the spring is ever in compression during the motion. (Hint: Try to determine the speed of the particle when the spring has its natural length.)

I get the energy of the system to be E= v^2 - 20x + 2(x-1)^2 for the first part. I also get E=-37 J when x=2 and v=1m/s. The problem is for b) when using the conservation of energy I get

v2 = -17

Does this mean I take the absolute value of v^2 and conclude the spring does compress as it has a velocity, or since square root of -17 gives a complex value do I conclude that this could not happen?
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May 5th, 2015, 06:33 PM   #2
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$E_{total}=\frac{1}{2}k(x-1)^2-mgx+\frac{1}{2}mv^2$

Quote:
(b) When the particle is 2 m below A, it has speed 1 m s−1
A? Do you mean O?

$E_{total}=2-40+1=-37 \, J$

When $x=1$, the gravitational potential energy is $-20 \, J$. Since no energy is stored in the spring at its natural length, the kinetic energy would have to be $-17 \, J$, but kinetic energy cannot be less than zero. Therefore, the mass can never get to the height of $1 \, m$ below O.

Have a look at the potential energy curve shown ... $f_1(x)=2(x-1)^2-20x$ is the sum of the elastic & gravitational potential energy; $f_2(x)=-37$ is the total mechanical energy of the system. Note the intersection of the two graphs defines the max and min position for the mass while it oscillates. The vertical distance between the two graphs is the kinetic energy of the mass, a max at $x=6 \, m$, where $mg=kx$.
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May 5th, 2015, 10:35 PM   #3
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Thanks very much skeeter that is a massive help. The question said A but I assume it is meant to say O! Thanks again
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May 6th, 2015, 02:06 AM   #4
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AJ235, please do not post duplicate topics.
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