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mjgillen08
Joined: 22 Feb 2010 Posts: 1 Location: Illinois

Posted: Mon Feb 22, 2010 11:27 pm Post subject: Hamonic Motion 


A 12kg body undergoes simple harmonic motion of amplitude 8.5cm and period .2 seconds.What is the magnitude of the maximum force acting on it?
Thanks for the help! 

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hepcj Site Admin
Joined: 23 Jun 2007 Posts: 125

Posted: Tue Feb 23, 2010 1:25 am Post subject: 


The maximum force occurs when there is the greatest acceleration.
The definition of SHM is that the displacement is in opposition to the acceleration. i.e. x =  k a
Maximum acceleration occurs when the body is at the fullest extent of travel and is just about to change direction.
From the mathematical description of SHM displacement, acceleration is the second derivative of the displacement:
displacement: x(t) = A sin(omega t)
velocity: x'(t) = A*omega cos (omega*t)
acceleration: x''(t) =  A*omega^2 sin(omega*t)
f =1/T = omega/(2pi)
therefore,
omega = 2pi/ T
at the end of the swing the sin(omega t) =1 so
accleration: x''(t) =A*omega^2 or A*(2pi/T)^2
m=12kg
A=8.5cm = 0.085 m (got to convert to metre so the units work out.)
T=0.2 sec
Substituting the above values into equation for acceleration:
magnitude of a = 0.085 * (2*pi/0.2)^2 = 83.89 ms^2
and from F= ma
F= 12* 83.89 = 1007 N 

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