Question

5, (3 points) A mass m 300 g lies on a frictionless horizontal surface and is attached to a horizontal spring with a spring constant k = 3 N/m. A coordinate system is given such that the z axis is parallel to the motion of the mass under the action of the spring, and the origin is located at the un-stretched position of the spring. The position of the mass is given by: x(t) = A cos(wt + φ) At time t = 2s, the position of the nnass m is x-0.2 m, and its velocity is u-_0.5 m/s. (a) Determine A, w, and φ, and make a plot of x(t) and u(t) between 0s and t = 10 s. (b) What is the mechanical energy of the mass? (c) What is the speed of the mass at position x = ?

Answer 1

W = squareroot k/m = squareroot 3/0.3 = 3.1635 ^-1 or 3.163 rad/s 0.2 = A cos(3.163 t + phi) + 0.5 = + A sin(3.163 t + phi) times 3/163 So, 0.2/0.5 = 1/3.163 cot (3.163 t + phi) Rightarrow cot (3.163 t + phi) = 1.265 tan (3.163 t + phi) = 0.79 Rightarrow 3.163 t + phi = 0.79 phi = 0.79 - 3.163 times 2 = -5.536 rad 0.2 = A cos(3.163 times 2 - 5.536) So, A = 0.284 m b) E = 1/2 k A^2 = 0.5 times 3 times (0.284)^2 = 0.121 J c) A/2 A cos (3.163 t - 5.536) t = 5.536 + 1.0472/3.163 Rightarrow t = 2.081 v = -0.284 times 3.163 sin (3.163 times 2.081 - 5.536) Rightarrow v = -0.7775 m/s