the answers are provided If the initial position of a block is x = .16 meters an

Question

the answers are provided

If the initial position of a block is x = .16 meters and its initial velocity is - 5 m/s, and K. = 100 N/m and M = 3 kg. What is the angular frequency omega , frequency f, period T? What is the amplitude A, phase angle theta ? Write the general equations for x, v, and a. What is the absolute value of the maximum values of x, v, a, K.E., U? omega = 5.77 rad/s, f = .92 Hz, T = 1.088 sec A = .88 meters, theta = 2.96 rads x = .88sin(5.77t + 2.63), v = 5.1cos(5.77t + 2,96), a = - 29.4sin(5.55t + 2.96) xmax = .88 meters vm = 5.1 m/s, am = 29.4 m/s, am = 29.4 m/s2, KEm = 38.7 joules, UM = 38.7 J

Explanation / Answer

Let the eqn be

x = A*sin(t+)

v = A*cos(t+)

at t= 0

x/v = (1/)*tan

We know that

= (K/M) = 5.77 rad/s

2.) a.) = 5.77 rad/s

f = /2 = 0.92 Hz

T = 2/ = 1.088 s

b.) 0.16 = A*sin()

-5 = A*5.77*cos()

=> tan = n (+- )tan-1(0.18464)

Also for above two eqn to be satisfied

sin > 0 & cos < 0

=> tan = - 0.1826 = 2.96 rad

A = 0.16/sin(2.96) = 0.88 m

c.) x = A*sin(t+) = 0.88*sin(5.77t + 2.96)

v = dx/dt = 0.88*5.77*cos(5.77t + 2.96) = 5.1*cos(5.77t + 2.96)

a = dv/dt = 5.1*5.77*-sin(5.77t + 2.96) = -29.4*sin(5.77t + 2.96)

d.) xmax = 0.88 m ( -1 < sin(5.77t + 2.96)< 1)

vmax = 5.1 m/s (-1< cos(5.77t + 2.96)<1)

amax = 29.4 m/s2 (-1 < sin(5.77t + 2.96) < 1)

K.E.max = (1/2)m*vmax2 = 0.5*3*(0.88*5.77)2 = 38.7 J

Umax = (1/2)kxmax2 = 38.7 J

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