The position of a particle is given by x = (7.0 cm) cos 6 pi t, where t is in se

Question

The position of a particle is given by x = (7.0 cm) cos 6 pi t, where t is in seconds. What are (a) the frequency. (b) the period, and (c) the amplitude of the particle's motion? (d) What is the first time after t = 0 that the particles is at its equilibrium position? In what direction is it moving at that time? Consider a mass, m, attached to a spring with spring constant k. The other end of the spring is attached to a motor that exerts a driving forte F(t) F_0 cos(omega t), where F_0 is a constant and omega is the driving frequency. When the motor is turned on the mass oscillates back and forth with the same frequency omega, but is also subject to a resistive (drag) force proportional to its velocity F_D = -m gamma v, where m is the mass and gamma is a constant. (a) Write clown the sum of the forces to determine the acceleration, x, of the mass. (b) By substitution, show that the equation you found in part (a) is solved by x(t) = A cos (omega t - delta), and determine the constants A and delta in terms of the natural spring frequency omega_0 driving frequency, omega, F_0, m, and gamma.

Explanation / Answer

1) a) frequency f = w/2pi = 6pi/2pi = 3Hz

b) period T = 1/f = 0.333 s

c)amplitude = 7cm

d) time t = 1/12 s, it is moving towards -x

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