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Mth 174
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A simple harmonic oscillator has restoring force constant k, mass m and experiences drag force -c v, where v is its velocity. What is the differential equation for the motion of the oscillator? What are the possible solutions to the equation?
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is that simple harmonic oscillator means no friction, d^2s/dt^2 = -k/m s, i don't understand the meaning of the drag force -c v, is this a simple harmonic model or damped harmonic model
@& A simple harmonic oscillator is one which is subject to a restoring force F = - k s.
That doesn't mean the F is the net force.
A typical restoring force F = - k s occurs for a mass on a spring, where s is the displacement from equilibrium.
If you attach a mass m to the spring then, if F = - k s is the net force, you get the equation
m s '' = - k s,
where s '' indicates the second derivative of s with respect to t (i.e., s '' is the acceleration; m s '' is mass * acceleration, which by Newton's Second Law is the net force).
Now if, let's say, you put the mass in water, then the water will tend to oppose the motion of the mass; the faster the mass moves through the water the greater the opposing force of the water. This force is called the drag force. It is also called a 'damping force' (not in the sense that water is wet and the mass will come out of the water damp, but in the sense that the opposing force tends to work against the motion; to work against something is to 'damp' it).
In any case, if the mass isn't moving too fast, the opposing force of the water might be proportional to the speed of the mass.
s ' is the velocity of the mass (the magnitude of s ' is the speed, and the direction of s ' is the direction of motion). The drag force might therefore be proportional to the speed but in the direction opposite the motion. Thus the drag force is
F_drag = - c s '.
Now the net force is
F_net = - k s - c s '
and the differential equation is
m s '' = - k s - c s '.
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