A mass of 100 g is attached to a spring and mounted for horizontal motion. It is

Question

A mass of 100 g is attached to a spring and mounted for horizontal motion. It is pulled 250 mm to the right of the position where the spring is unstretched and released. The resulting free vibrations, which are damped by friction, have a frequency of 0.4 Hz. It is observed that the first motion to the right takes the mass to a point 15 nun to the left of the starting point. Determine the coefficient of resistance for the system and the natural frequency this system would have in the absence of damping. (Hint: Draw a diagram of the problem.)

Explanation / Answer

The equation of motion for a damped harmonic motion is m * a = -k * x - b * v a = (d2x/dt2) and v = (dx/dt) or m * (d2x/dt2) = -k * x - b *(dx/dt) or m * (d2x/dt2) + b * (dx/dt) + k * x =0 The solution of the above equation is x = A * e-t * cos(w't) -----------(1) where A,,and w' are assumed to be constants,and x = A att = 0. x = 15 mm,A = 250 mm and w' = 2f' where f' = 0.4Hz solving equation (1) for we get = (1/t) * exp[(A/x) * cos(w't)] = (b/2m) b is damping constant or b = 2m * m = 100 g = 100 * 10-3 kg The natural frequency this system would have in the absence ofdamping is w = (k/m)1/2 -----------(2) F = k * x or m * g = k * x or k = (m * g/x) ------------(3) from (2) and (3) w = ((m * g/x)/m)1/2 or w = (g/x)1/2 g = 9.8 m/s2 g = 9.8 m/s2

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