In the system shown in. the figure below, m - 500 gm and k - 0.1 kN/m. Assume ac

Question

In the system shown in. the figure below, m - 500 gm and k - 0.1 kN/m. Assume acceleration due to gravity g - 9.8 m/s2. When the mass m is at rest, by how much is the spring stretched? Using the Free Body Diagram (FBD) of the mass, determine the differential equation governing the vertical motion of the mass. Suppose the mass is pulled down vertically from its rest position by 1 cm and then released. Determine the ensuing motion of the mass along the vertical. Suppose at its rest position, the mass is suddenly imparted a vertically downward impulse of 1 Ns. Determine the ensuing motion of the mass along the vertical. You can assume the duration of the impulse to be infinitesimally small. Hint: For parts c.) and d.), solve using Laplace transforms the governing differential equation for the vertical motion of the mass together with appropriate initial conditions.

Explanation / Answer

a) x = mg/k = 4.9 cm

b) ma = mg - kx
=> a = g - kx/m
=> d2x/dt2 = g - kx/m

c) y = cos(t) + mg/k

= (k/m) = (100/.5) = 102

so: y = cos(102t) + 4.9

d) v = I/m = 1/.5 = 2 m/s

so:

y = .144sin(102t) + 4.9

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