Physics Assignment Question 2 Use Newton’s second law to determine the equation of motion for the simple pendulum with mass M and length L. Use the small angle approximation to show that this equation can be expressed in the form of the equation of motion for simple harmonic motion. Based on this simple harmonic motion equation, what is the angular frequency ?? Question 3 A damped harmonic oscillator is aligned on the y axis, with Hooke’s law Fs = ?ky and damping force Fd = ?bvy. A mass m is attached to the end of the ideal spring. The natural frequency is given by ?0 = ? k m . 1. Use Newton’s second law to determine the equation of motion for this damped harmonic oscillator. 2. Determine the characteristic equation using test solution y = ert, and show how in the roots of this equation appears the angular frequency ? = ? ?0 ? ?2 4 . 3. Express conditions for ? and ?0 such that the angular frequency ? is: a) real, b) zero, c) imaginary. Name what damping type each of these conditions correspond to, and make a rough sketch of y(t) vs. t for each. 4. For the light damping case, where y(t) = Ae??t/2 cos (?t + ?), show that this function is indeed a solution to the equation of motion you wrote down in Part 1. Question 4 For the damped harmonic oscillator in Question 3, write down the same equation of motion with a forcing term. Explain using words and equations the difference between an undamped and damped forced oscillator, focusing on how the amplitude of oscillations change in the two cases. Discuss the expected amplitude and driving fre- quency during resonance in the two cases.