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Transcript

A mass of 25g is attached to a vertical spring with a spring constant k = 3 dyne/cm. The surrounding medium has a damping constant of 10 dyne*sec/cm. The mass is pushed 5 cm above its equilibrium position and released. Find (a) the position function of the mass, (b) the period of the vibration, and (c) the frequency of the vibration. x = position of the block v = x' = velocity of the block m = mass of the block = 25 g = 0.025 kg k = spring constant = 3 dyne / cm = 3 ×10−5 N / 10-2 m = 3 ×10−3 N / m B = damping constant = 10 dyne ·s / cm = 10×10−5 N· s/10-2 m = 10−2 N· s / m Since we adjust the coordinate system so that x = 0 corresponds to the spring being unstretched, then the stretch of the spring is simply equal to x. The spring force becomes Fspring = − k x In addition, there is a damping (friction) force that resists the motion. It is proportional to the velocity. So we add Fdamping = −B v to get the total force F = Fspring + Fdamping = − k x − B v Combining this with Newton's law of motion F = m a, and the definition of acceleration as the second derivative of position a = x'' we have the differential equation: m x'' = −k x − b v or equivalently: To simply the equation, we define the following parameters: 0 k 3 103 N / m 3 rad / s m 0.025kg 5 and B 102 N s / m 0.2rad / s 2m 2 0.025kg The differential equation now becomes: x 2 x 02 x 0 Continuing, we can solve the equation by assuming x Ae t Substituting this assumed solution back into the differential equation, we obtain: 2 2 02 0 Solving for , we find: 2 02 Since 2 02 0 , is complex, the system is under-damped. 2 02 i 02 2 Substituting back to the assumed solution x Ae t x Ae 2 02 t Ae t e i 02 2 t Ae t cos( 02 2 t ) iAe t sin( 02 2 t ) If we only take the real part, x(t ) Ae t cos(d t ) , where d 02 2 ( 3 2 ) (0.2) 2 0.2 2 0.283rad / s 5 At the initial point t =0, the mass is 5 cm above its equilibrium. x(0) A 0.05m So (a) x(t ) 0.05e0.2t cos(0.283t ) , the positive x direction is downward. (b) The frequency of the vibration is f (c) The period of the vibration is T d 0.283rad / s 0.045Hz 2 2 1 1 22.214s f 0.045Hz