The Physical Pendulum Damped Oscillations Forced Oscillations Chapter 12 – part C The Physical Pendulum Damped Oscillations Forced Oscillations

In-phase and quadrature components of motion     Interesting properties:            

The forced and damped harmonic oscillator       For X:…   For Y:…   From the last one and considering the relation between sin and cos:    

From the X equation, one can work out A:   Replacing cos j and sin j:  

Exercise 12.28 28. A very light rigid rod with a length of 0.500 m extends straight out from one end of a meter stick. The stick is suspended from a pivot at the far end of the rod and is set into oscillation. (a) Determine the period of oscillation. (b) By what percentage does the period differ from the period of a simple pendulum 1.00 m long?

Exercise 12.31 31. A pendulum with a length of 1.00 m is released from an initial angle of 15.0°. After 1 000 s, its amplitude has been reduced by friction to 5.50°. What is the value of b/2m?

Exercise 12.31 33. A 2.00-kg object attached to a spring moves without friction and is driven by an external force F = (3.00 N) sin(2πt). Assuming that the force constant of the spring is 20.0 N/m, determine (a) the period and (b) the amplitude of the motion.

Exercise 12.24 24. The angular position of a pendulum is represented by the equation θ = (0.032 0 rad) cos ωt, where θ is in radians and ω = 4.43 rad/s. Determine the period and length of the pendulum.

Exercise 12.36 36. Damping is negligible for a 0.150-kg object hanging from a light 6.30-N/m spring. A sinusoidal force with an amplitude of 1.70 N drives the system. At what frequency will the force make the object vibrate with an amplitude of 0.440 m?

Exercise 12.49 49. A horizontal plank of mass m and length L is pivoted at one end. The plank’s other end is supported by a spring of force constant k. The moment of inertia of the plank about the pivot is . The plank is displaced by a small angle θ from its horizontal equilibrium position and released. (a) Show that it moves with simple harmonic motion with an angular frequency . (b) Evaluate the frequency, assuming that the mass is 5.00 kg and that the spring has a force constant of 100 N/m.