The graph below describes a 175g mass vertically hung from a spring of negligibl

Question

The graph below describes a 175g mass vertically hung from a spring of negligible mass.

a) What is period of motion?

b) What is the spring constant of the spring?

c) How much work was required to stretch the spring?

d) What is the phase shift of the position function?

e) Write an expression for the velocity of this object as a function of time?

f) What is the acceleration of the object when t = 0.1 seconds?

The graph below describes a 175g mass vertically hung from a spring of negligible mass. What is period of motion? What is the spring constant of the spring? How much work was required to stretch the spring? What is the phase shift of the position function? Write an expression for the velocity of this object as a function of time? What is the acceleration of the object when t = 0.1 seconds?

Explanation / Answer

a)

Period = T = 0.117 sec

b)

T = 2*pi*sqrt(m/k)

so,

k = m*4*pi^2/T^2 = 0.175*4*3.14^2/0.117^2 = 504.18 N/m

c)

A = amplitude = 0.25 m

work required = 1/2*k*A^2 = 1/2*504.18*0.25^2 =15.75 J

d)

we have, SHM equation as

x = A*cos(w*t + phase)

at t=0 ; x= 0.155 m

so,

0.155 = 0.25*cos(phase) ...

phase = arccos(0.155/0.25) = 51.68 degree = 0.902 rad

e)

w = sqrt(k/m) = sqrt(504.18/0.175) = 53.675 rad/s

x = A*cos(w*t + phase)

V= dx/dt = -Aw*sin(w*t + phase)

= -0.25*53.675*sin(5.23*t+0.902)

= -13.42*sin(53.675*t+0.902)

f)

a = dV/dt = -13.42*53.675*cos(53.675*t+0.902)

at t=0.1

a = -13.42*53.675*cos(53.675*0.1+0.902) = -720.25 m/s^2

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