# A spring mass system is characterized by 1 16Nm− k = and m = 0.1 kg.The system is oscillating with an amplitude of 0.20 m. i) Calculate the angular frequency of oscillation. ii) Obtain an expression for the velocity of the block as a function of displacement and calculate its value at x = 1.0 m. iii) Also calculate energy of the spring-mass system.

1
by aarzu

2015-04-13T13:15:40+05:30

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equation of motion and force:
F = m a = m d²x/dt² = - k x
d²x/dt² = - k/m x

let x = A Sin (ωt + Ф)
then  d²x/dt² = - A ω² Sin(ωt+Ф) = - ω² x

k = 116 N/m  or  16 N/m  ???      which one ?
m = 0.1 kg
A = 0.20 meters

SHM :  angular frequency = ω = √(k/m) = √(116/0.1) = 34 rad/sec
if  k = 16 N/m,    ω = √160 = 4√10 rad/sec
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x = A Sin (ωt + Ф)          = displacement from the mean position
v = velocity of the particle executing the SHM
v = dx/dt = A ω Cos (ω t + Ф)
v  = ω  √A² - x² )  = ω A  √[1 - x²/A² ]
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x = 1.0 m      this value is not possible, as  amplitude is 0.20 m.  SO x has to be less than 0.2 m.  Is it 0.1 meters ?      x has to be less than or equal to amplitude.

v = 4√10 rad/sec * √[0.2² - 0.1²] meters = 2.19 m/sec

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Energy of the spring mass system :
= 1/2 m v² at the mean equilibrium position , as x = 0  and PE = 0
= 1/2 k A²  at the extreme position when x = A, as  v = 0 and KE = 0.
= 1/2 * 0.1 kg * 0.20² = 0.002 Joules