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bobred
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1. Find the equation for conservation of energy in system. System consists of three springs A,B and C with stiffness k, 2k and 0.5k respectively and natural lengths l, 0.5l and 2l respectively.
Equation for conservation of energy E=K+U_{spring}+U_{gravity}
K=\frac{1}{2}mv^2
Datum taken at point A, U_{gravity}=-mgx
U_{spring}=\frac{1}{2}k(x-l_0)^2
Where K is kinetic energy and k is the spring stiffness and l_0 is the natural length.
By analysing the diagam
U_{springA}=\frac{1}{2}k(x-l_0)^2
U_{springB}=\frac{1}{2}*2k(x-\frac{1}{2}l_0-\frac{1}{2}l_0)^2=k(x-l_0)^2
U_{springC}=\frac{1}{2}*\frac{1}{2}k(4l_0-x-2l_0)^2=\frac{1}{4}k(2l_0-x)^2
K=\frac{1}{2}mv^2
U_{gravity}=-mgx
My question is, is the answer E=K+U_{gravity}+U_{springA}+U_{springB}+U_{springC} and have I derived the individual energies correctly?
Homework Equations
Equation for conservation of energy E=K+U_{spring}+U_{gravity}
K=\frac{1}{2}mv^2
Datum taken at point A, U_{gravity}=-mgx
U_{spring}=\frac{1}{2}k(x-l_0)^2
Where K is kinetic energy and k is the spring stiffness and l_0 is the natural length.
The Attempt at a Solution
By analysing the diagam
U_{springA}=\frac{1}{2}k(x-l_0)^2
U_{springB}=\frac{1}{2}*2k(x-\frac{1}{2}l_0-\frac{1}{2}l_0)^2=k(x-l_0)^2
U_{springC}=\frac{1}{2}*\frac{1}{2}k(4l_0-x-2l_0)^2=\frac{1}{4}k(2l_0-x)^2
K=\frac{1}{2}mv^2
U_{gravity}=-mgx
My question is, is the answer E=K+U_{gravity}+U_{springA}+U_{springB}+U_{springC} and have I derived the individual energies correctly?
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