A 0.2 kg block connected to a spring oscillates as shown in the graph below a) f

Question

A 0.2 kg block connected to a spring oscillates as shown in the graph below

a) find the amplitude of motion

b) find the period of motion

c) calculate the angular frequency ( as a multiple of ? )

d) calculate the spring constant

e) determine the initial phase (consider the displacement at t=0 s to be x = 0.125 m)

f) write the expression for the displacement as a function of time

g) write the expression for velocity as a function of time

h) write the expression for acceleration as a function of time

A 0.2 kg block connected to a spring oscillates as shown in the graph below find the amplitude of motion find the period of motion calculate the angular frequency ( as a multiple of ? ) calculate the spring constant determine the initial phase (consider the displacement at t=0 s to be x = 0.125 m) write the expression for the displacement as a function of time write the expression for velocity as a function of time write the expression for acceleration as a function of time

Explanation / Answer

a) Amplitude = max. displacement from mean = 0.25 m

b) Period = duration to copmlete 1 repetation of motion = T = 12 s

c) angular frequency= w = 2*pi/T = 2*3.14/12 = 0.5233 s^-1

d) k = w^2*m = 0.5233^2*0.2 = 0.055 N/m

e)

let phi = phase

x = A*sin(pi*wt+phi)

at t=0 s ; x = 0.125 m

so,

0.125 = 0.25*sin(phi)

phi = arcsin(0.125/0.25) =0.523598776 rad = 30 degree

f)

x = A*sin(pi*wt+phi)

x = 0.25*sin(3.14*0.5233*t+30) = 0.25*sin(1.64*t+30)

g)

V = dx/dt = d/dt(0.25*sin(1.64*t+30)) = 0.25*1.64*cos(1.64*t+30)) = 0.41*cos(1.64*t+30)

V=0.41*cos(1.64*t+30)

h)

a=dV/dt = d/dt(0.41*cos(1.64*t+30))) = -0.41*1.64*sin(1.64*t+30))) = -0.67*sin(1.64*t+30)

a = -0.67*sin(1.64*t+30)

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