An object of mass 2 grams is attached to a vertical spring with spring constant 32 grams/sec“. Neglect any friction with the air. (a) Find the differential equation y" = f(y, y') satisfied by the function y, the displacement of the object from its equilibrium position, positive downwards. Write y for y(t) and yp for y' (t). y" = Σ (b) Find r1, r2, roots of the characteristic polynomial of the equation above. r1, r2 = Σ (b) Find a set of real-valued fundamental solutions to the differential equation above. Y1 (t) = Σ Y2(t) = Σ (c) At t = 0 the object is pulled down v2 cm and the released with an initial velocity upwards of 4/2 cm/sec. Find the amplitude A > 0 and the phase shift o E (-1, 7] of the subsequent movement. A = ΣΦ Σ

A vector a of magnitude 17 units and another vector b of magnitude 8.5 units differ in directions by 72°. Find (a) the scalar product of the two vectors and (b) the magnitude of the vector product a×b.

Answer must be in standard form scientific notation with SI units that do not have prefixes ( except for kg.) all angles and directions must be calculated to the nearest 0.1 deg. (Tenth of a degree) and entered in standard form scientific notation.

1. ANALYTICAL METHOD OF ADDING VECTORS Solve the same problem as PART 1 but this time analytically. START with the values below. DO NOT USE VALUES FROM PART 1. Suppose you first walk A = 12 m in a direction 20° west of north and then B= 20 m in a direction 40° south of west. Let the +x-axis coincide with the easterly direction and the +y-axis with the northerly direction. Note: In this problem, do not round values if you need to use them for calculations in subsequent parts. Rounding errors can propagate and can make your final answer lie outside the tolerance limit of 5% set for this problem. Keep all answers to 4 significant figures.