A 0.25-kg mass at the end of a spring oscillates 2.2 times per second with an amplitude of 0.15 m. Determine

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. The maximum velocity is 2π times mmplitude times frequency and that's given by equation 11-7. So, that's 2π times the amplitude of 0.15 meters times the frequency of 2.2 hertz which gives 2.1 m/s maximum speed. The speed at any position is the maximum speed times square root of 1 minus the position squared divided by the amplitude squared, that's equation 11-5B. So, that's maximum speed taken from part A 2.0735 meters per second times square root of 1 minus 0.1 meter squared divided by 0.15 meters squared. And that gives 1.5 meters per second is the speed at position of 0.1 meters the total energy is 1/2 mass times the maximum speed squared, that's 1/2 times 0.25 kilograms times 2.0735 meters per second squared. And that gives 0.54 joules. And the equation for its position as a function of time is using, can be using cosine since we're told that at time equal to 0 we have x is the maximum, and that's true for cos function. Cos function looks like this, where you have at x equals 0 we have at its maximum of 1. So, amplitude times cos of ωt is going to be the position as a function of time and ω is at 2π f. And so we have 0.15 meters for amplitude times cosine of 2π times 2.2 hertz for frequency times time and 2 times 2.2 is 4.4. So, we have function of time, the position is 0.15 meters times cos of 4.4π times t.