An object with mass 2.7 kg is executing simple harmonic motion, attached to a spring with spring constant k = 310 N/m. When the object is 0.020 m from its equilibrium position, it is moving with a speed of 0.55 m/s.

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. Total energy is 1/2 m times velocity squared plus 1/2 kx squared. This is kinetic energy plus potential energy. And that total energy will equal the potential energy when the object is at its maximum displacement at which point it has zero velocity. So, this total energy is 1/2 k A squared. And we can solve this for A. And we'll, you know, divide everything by 1/2 or multiply both sides by 2 whichever way you want to look at it that gets rid of the halves. And divide both sides by k. And you're left with m over k times v squared plus x squared equals A squared. Then take the square root of both sides and we have amplitude is the square root of mass over k times v squared plus x squared, all square rooted. That's 2.7 kilograms times 310 newtons per meter times 0.55 meters per second squared plus 0.02 meters squared. And that gives 0.055 meters for the amplitude. And doing a reality check on that number, it should be greater than x because it can never reach a displacement from equilibrium greater than the amplitude, and this number is greater than 0.02. So, the answer could be right. And it's 1/2 m Vmax squared is equals 1/2 k A squared. This is the total energy expressed in terms of kinetic energy when it's at the equilibrium position, that's when it has v max, at which point it has no potential energy. And that equals the total potential energy when it's at its maximum displacement at which point it has no speed. And we can solve this for v max and divide both sides by m and multiply both sides by 2. And you have v max after taking the square root of both sides is square root of k over m times A. That's square root 310 newtons per meter over 2.7 kilograms times 0.05509 meters, which is 0.59 meters per second.