Question

6. A mass on a spring has an angular oscillation frequency of 2.56 rad/s. The spring constant is 27.2 N/m and the system’s kinetic energy is 5.03 J. When t = 1.56 s. (a) What is the velocity of the mass at t = 1.56 s. (b) What is the oscillation amplitude? assume that its maximum displacement occurs at t=0 s and θ = 0.

Answer 1

  Kinetic energy, KE = 1/2*mv^2.

For a spring system, the the velocity is obtained from the position function:
x(t) = A cos(wt), where,
-- x(t) is position as function of time
-- A is amplitude
-- w (greek omega) is angular frequency
-- t is time

We know velocity is the derivative of position, so v = dx/dt = - A*w sin(wt).
Thus, v^2 = A^2 * w^2 * sin^2(wt)

We can then relate all of this with the kinetic energy equation, KE = 1/2*m*v^2, so
KE = 1/2* m* A^2 * w^2 * sin^2(wt).

Now in your problem, you don't know the mass of the block. But there is a little trick around this. The spring constant k = mw^2 (you can derive this using F=ma). So, you can replace (m*w^2) with k in the equation.

So the final equation you are looking for to relate those three variables is:
KE = 1/2 * k * A^2 * sin^2(wt).

You can simply solve for A since you know all four other variables. And be sure you use radians when computing the sine function.

A = 0.702 mts