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A spring stretches 0.150 m when a 0.30 kg mass is hung from it. The spring is then stretched an additional 0.100 m from this equilibrium point and released. Determine
a) the spring constant
b) the maximum velocity of the mass
c) the maximum acceleration of the mass
d) the total mechanical energy of the mass
e) the period and frequency of the mass and spring and
f) the equation of time-dependent vertical position of the mass

Sagot :

okpalawalter8

Answer:

a)  [tex]k=19.6N/m[/tex]

b)  [tex]V_m=0.81m/s[/tex]

c)  [tex]a_m=6.561m/s^2[/tex]

d)  [tex]K.E=0.096J[/tex]

e)  [tex]T=0.78sec[/tex] &[tex]F=1.29sec[/tex]

f)   [tex]mx'' + kx' =0[/tex]

Explanation:

From the question we are told that:

Stretch Length [tex]L=0.150m[/tex]

Mass [tex]m=0.30kg[/tex]

Total stretch length[tex]L_t=0.150+0.100=>0.25[/tex]

a)

Generally the equation for Force F on the spring is mathematically given by

[tex]F=-km\\\\k=F/m\\\\k=\frac{m*g}{x}\\\\k=\frac{0.30*9.8}{0.15}[/tex]

[tex]k=19.6N/m[/tex]

b)Generally the equation for Max Velocity of Mass on the spring is mathematically given by

[tex]V_m=A\omega[/tex]

Where

A=Amplitude

[tex]A=0.100m[/tex]

And

[tex]\omega=angulat Velocity\\\\\omega=\sqrt{\frac{k}{m}}\\\\\omega=\sqrt{\frac{19.6}{0.3}}\\\\\omega=8.1rad/s[/tex]

Therefore

[tex]V_m=A\omega\\\\V_m=8.1*0.1[/tex]

[tex]V_m=0.81m/s[/tex]

c)

Generally the equation for Max Acceleration of Mass on the spring is mathematically given by

[tex]a_m=\omega^2A[/tex]

[tex]a_m=8.1^2*0.1[/tex]

[tex]a_m=6.561m/s^2[/tex]

d)

Generally the equation for Total mechanical energy of Mass on the spring is mathematically given by

[tex]K.E=\frac{1}{2}mv^2[/tex]

[tex]K.E=\frac{1}{2}*0.3*0.8^2[/tex]

[tex]K.E=0.096J[/tex]

e)

Generally the equation for  the period T is mathematically given by

[tex]\omega=\frac{2\pi}{T}[/tex]

[tex]T=\frac{2*3.142}{8.1}[/tex]

[tex]T=0.78sec[/tex]

Generally the equation for  the Frequency is mathematically given by

[tex]F=\frac{1}{T}[/tex]

[tex]F=1.29sec[/tex]

f)

Generally the Equation of time-dependent vertical position of the mass is mathematically given by

[tex]mx'' + kx' =0[/tex]

Where

'= signify order of differentiation

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