For a mass on a spring, or a pendulum oscillating near its equilibrium position, we postulate a 'drag' force proportional to the velocity of the pendulum.

• The resulting equation for the net force on the mass is as indicated the first line below.

• - c v indicates the drag force, which is proportional to and in the opposite direction to the velocity of the mass.
• - k x indicates the usual linear restoring force.

• Since F = m a = m d^2 s / dt^2 and v = dx / dt, the equation can be expressed as in the second line.

To solve this equation we use the following strategy:

• We place all nonzero terms on the left-hand side of the equation.
• We use the trial solution x = C e ^ (rt).
• Substituting the derivatives of this function into the equation and dividing through by C e ^ (rt), we obtain the values of r indicated in the last line.

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Provided the discriminant (the quantity under the square root sign) is positive, we obtain two real solutions r1 and r2.

• The discriminant will be positive if c is relatively large compared to k * m (e.g., for a small mass in a very thick fluid).
• In this case any constants C1 and C2 yield a solution x = C1 d^(r1 t) + C2 e^(r2 t).
• In the event that k * m is close to c^2, the discriminant will be close to 0 and our two values r1 and r2 will be very nearly the same. In this case we have something very close to an exponential decay toward the equilibrium position.

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If c is small, corresponding to a relatively small drag, then the discriminant is negative and its square root is imaginary.

• In that case r1 and r2 will be complex numbers.
• It will follow that e ^ (rt) will have as a factor an exponential function with an imaginary exponent.

Before we deal with this case we consider what an exponential function of an imaginary quantity might involve.

• Using the Taylor expansion of e^x, we obtain the Taylor expansion of e^( i * `theta ), where i stands for `sqrt(-1).
• A little algebra shows us that e ^ (i `theta) = cos(`theta) + i sin(`theta).

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The detailed solution, with quantities in exponential form, is indicated below.

We note that r1 and r2 are -a +- b i, where a = -c / (2m) and b = `sqrt( 4 k m - c^2) / (2m).

The resulting exponential functions are each factored into a real part e ^ -(at) and the complex part ( C1 e ^ (ibt) + C2 e ^ (-ibt) ).

The complex part of the solution is expressed according to Euler's formula, then the like terms involving the cosine and sine functions are collected.

Since C1 and C2 can be any complex numbers, (C1 + C2) and (C1 - C2) can be any complex numbers.

• If we make the substitutions C1 = A/2 - i B / 2, C2 = A / 2 + i B / 2, we obtain C1 + C2 = A and C1 - C2 = - i B.
• This yields the solution in the last line, where x(t) is a linear combination of e^(-at) cos(bt) and e^(-at) sin(bt).

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Using trigonometric identities the same solution can be expressed in the form C e ^ (-at) sin ( b ( t - t0) ).

• The sine function is characterized by its angular frequency b = `sqrt(k m - c^2) / (2m) and by its horizontal shift t0. The graph of this function is indicated.
• Multiplying the sine function by the exponential function gives us the graph depicted at the bottom of the figure below.

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The preceding case, in which the position of the mass is represented by a sine function with an exponentially decaying amplitude, is called the 'underdamped' case.

• We say that the drag force 'damps' the motion of the mass.
• If the drag constant c is not too large compared to the product k * m of the object's mass and the force constant, then the object will be able to oscillate back and forth, though with a decreasing amplitude.
• The damping is therefore not as great as in the case where the object more or less exponentially approaches its equilibrium position without oscillating back and forth.

The case where the drag is very large is therefore called 'overdamped'.

• The difference between the overdamped and underdamped case is the difference between a negative and a positive discriminant in the quadratic formula.

When the discriminant is 0, we are right at the border between the overdamped and underdamped case. This case is called 'critically damped'.

Typical graphs of the overdamped and underdamped cases are indicated below.

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