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Consider a mass m on the end of a spring of force constant
Chapter 4, Problem 4.28(choose chapter or problem)
Consider a mass m on the end of a spring of force constant k and constrained to move along the horizontal x axis. If we place the origin at the spring’s equilibrium position, the potential energy is \(\frac{1}{2} k x^{2}\). At time \(t=0\) the mass is sitting at the origin and is given a sudden kick to the right so that it moves out to a maximum displacement at \(x_{\max }=A\) and then continues to oscillate about the origin.
(a) Write down the equation for conservation of energy and solve it to give the mass’s velocity \(\dot{x}\) in terms of the position x and the total energy E.
(b) Show that \(E=\frac{1}{2} k A^{2},\) and use this to eliminate E from your expression for \(\dot{x}\). Us the result of (4.58), \(t=\int d x^{\prime} / \dot{x}\left(x^{\prime}\right)\), to find the time for the mass to move from the origin out to a position x.
(c) Solve the result of part (b) to give x as a function of t and show that the mass executes simple harmonic motion with period \(2 \pi \sqrt{m / k}\).
Questions & Answers
(1 Reviews)
QUESTION:
Consider a mass m on the end of a spring of force constant k and constrained to move along the horizontal x axis. If we place the origin at the spring’s equilibrium position, the potential energy is \(\frac{1}{2} k x^{2}\). At time \(t=0\) the mass is sitting at the origin and is given a sudden kick to the right so that it moves out to a maximum displacement at \(x_{\max }=A\) and then continues to oscillate about the origin.
(a) Write down the equation for conservation of energy and solve it to give the mass’s velocity \(\dot{x}\) in terms of the position x and the total energy E.
(b) Show that \(E=\frac{1}{2} k A^{2},\) and use this to eliminate E from your expression for \(\dot{x}\). Us the result of (4.58), \(t=\int d x^{\prime} / \dot{x}\left(x^{\prime}\right)\), to find the time for the mass to move from the origin out to a position x.
(c) Solve the result of part (b) to give x as a function of t and show that the mass executes simple harmonic motion with period \(2 \pi \sqrt{m / k}\).
ANSWER:Step 1 of 4
(a)
Kinetic energy;
\(K=\frac{1}{2} m \dot{x}^{2}\)
Potential energy;
\(U=\frac{1}{2} k x^{2}\)
According to the conservation, the energy can neither be created nor be destroyed.
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