An object of mass 306 g is attached to the base of a spring, with spring constan

Question

An object of mass 306 g is attached to the base of a spring, with spring constant 25 N/m, that is hanging from the ceiling. A pen is attached to the back of the object, so that it can write on a paper placed behind the mass-spring system. Ignore friction. a) Describe the pattern traced on the paper if the object is held at the point where the spring is relaxed and then released at t=0. The answer is "a vertical straight line of length 24 cm" I just do not know why. b) The experiment is repeated, but now the paper moves to the left at constant speed as the pen writes on it. Sketch the pattern traced on the paper. Imagine that the paper is long enough that it doesn't run out for several oscillations. The answer is "a positive cosine plot of amplitude 12 cm" I just do not know why.

Explanation / Answer

In case A, you will observe a straight vertical line drawn by the pen on the bouncing mass. It will descend to a distance of dmax = 2*m*g/k, or in our case, dmax = 23.99 cm. In case B, you will observe a cosine wave traced out on the moving paper. Its amplitude will be A = m*g/k, and its wavelength will be lambda = 2*Pi*v*sqrt(m/k), where v is the paper speed, m is the oscillating mass, k is the spring constant, and g is the gravitational field strength. It will trace the shape of y = m*g/k*cos(x*sqrt(k/m)/v) - m*g/k

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