A mass m is held by a spring with spring constant k. The mass is immersed in a c

Question

A mass m is held by a spring with spring constant k. The mass is immersed in a cup of

water (see the figure). The water exerts a viscous force ?bv on the mass; v is the velocity of the mass relative to the liquid, and b is a positive constant. A indicates the position of the suspension point of the spring, B the equilibrium position of the mass m, and C the position of the bottom of the cup.

a. The mass is displaced vertically from its equilibrium (B); it is then released. Find the differential equation of vertical motion of the mass m. The cup is at rest.

The cup is now moved up and down at an angular frequency ?. The position of the bottom of the cup (C) is given by d1(t) = D1 cos(wt).

b. Find the differential equation for the position x(t) of the mass m. Give your answer in terms of m, k, b, D1, and ?. Remember that v in the viscous force is the velocity relative to the liquid.

c. What is the steady state amplitude of the mass m? Give your answer in terms of m, k, b, D1, and ?.

In addition to driving the cup, we now also drive the mass by moving the suspension end (A) of the spring up and down with the same frequency w. The position of the suspension end is given by d2(t) = D2 cos(wt + phi).

d. Write down the differential equation for x(t) in the case that both the cup and the spring are driven.

e. Find D2 and phi, for which the steady state solution is x(t) = 0 at all times when both the cup and the spring are driven and the spring are driven.

Explanation / Answer

1) m * dv/dt = b * dx/dt

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