Question

1) Amass, m, on a spring with spring constant k obeys the equation of motion Where-1 kg. Andk is assigned a value 1 (in Sl units) What are the units of the spring constant? Assuming that at time O, the mass mis at rO traveling with a velocity of 1 m/s Work out the maximum displacement of the mass in subsequent oscillations Can you find an alternative way of getting this answer? 2) Amass,, on a spring with spring constant k obeys the equation of motion Showing your working establish that x = Acos(ult + g) is a solution. Also show that x = B sin(art + y) is a soltion. For both solutions to be viable there must be a relationship between them. a. What is the relationship between (A, B)? b. Give the relationship betweeny2 c. Finally, how are φ and γ related 3) Using the exponential form of sin and cos show that sin2 θ + cos2 θ-1 4) Using the series expansion of sin and cos obtain estimates for sinr and cos r when x is small 5) A disk is rotating in 2D with an argular frequency 6 radians per second. How many complete rotations does the disk male per second? The x, y coordinates of the disk arex cost, ysin t Draw a sketch showing the disk, the coordinates system and coordinates What is the value of ω? Which direction is the disk rotating in? Why? 6) Twowaves of equal amplitude are superimposed. Assume the relatve phase s n.and their amplitudes are equal. What is the resultant? Keeping the amplitudes and phases the same One wave is now adjusted to frequency of 25 Hz the other to 27 Hz. Using the excel file cven in lectures, or otherwise, explain what you would hear if you could listen to the two waves superimposed? Sketch the broad features of the superposed waves. 7) Discuss (100 words or less), why you think complex numbers can be used in the description of physical systems. Give one useful feature of doing so.

Answer 1

F = m d^2x/dt^2 = -k_1 x m = 1 kg k_1 = 1 f = -k_1 x n = - k_1 times m k_1 = newton/meter unit of force constant k_1 = n/m condition are at t = 0 x = 0 v/ = m/s m d^2 x/dt^2 = - k_1 x here m = 1 kg, k_1 = 1 n/m so, d^2 x/dt^2 = - x rightarrow d^2 x/dt^2 + x = 0 solution for this 2^nd order homogeneous equation is d^2 + 1 = 0 d = plusminus I so, x = e^0t [a cos t + b isn t] x = a cos t + b sin t at x = 0, t = 0 0 = a cos (0) + b sin (0) rightarrow a = 0 dx/dt = - a sin t + b cos t v = 0