Pull both pendulum bobs to the side (at the same displacement) and release them. The effect of drag on the larger, less massive orange bob is greater than that on the red bob, and its motion decays much more quickly than that of the red bob, for which the damping is much less noticeable.

Suspended so that their centers of mass are 1.0 m from the top end of their strings are a steel bob (red) that has a diameter of two inches (5 cm) and a mass of about 540 g, and a plastic ball (orange, made to resemble a basketball) whose diameter is 3.5 inches (8.9 cm) and whose mass is about 30 g. These constitute simple pendulums that have the same period (and frequency), but for which the effects of damping due to air drag, which we will address below, are greatly different. The force on the pendulum bob is that of gravity, F = mg, which is balanced by the tension in the string on which it hangs. If you displace the pendulum to one side, this force is resolved into two components. The vertical component is mg cos θ, and the horizontal component is mg sin θ, where θ is the angle through which you have displaced the pendulum. Gravity thus provides a restoring force, F = -mg sin θ, to bring the pendulum bob back to its equilibrium position.  For small angles, sin θ ≈ θ, so the horizontal displacement, x, approximately equals lθ, and the restoring force is F = -(mg/l)x. This leads to the differential equation

m(d²x/dt²) + (mg/l)x = 0.

Except for the constant in the x term, this is identical to the equation of motion for the mass-spring system given in the page for 40.12 -- Mass-springs with different spring constants and masses. As is shown on that page, this equation yields a solution

x = x₀ cos (ωt - φ),

where x₀ ≈ lθ_max, where θ_max is the angle to which you have displaced the pendulum bob from its equilibrium position (and thus its maximum angular displacement), and ω = √(g/l). The frequency at which the pendulum oscillates, in cycles per second, is ν = ω/2π, and the period, T, equals 2π√(l/g). As noted on the page for demonstration 40.21 -- Pendulums of different lengths and masses, we see from this that the frequency of oscillation is inversely proportional to the square root of the length of the pendulum, the period is proportional to the square root of the length of the pendulum, and both are independent of the mass of the bob.

Both pendulum bobs experience drag as they swing through the air. Since the diameter of the orange ball is 1.75 times that of the red bob, the drag force on it is about three times that on the red bob. Because the mass of the red bob is about 18 times that of the orange ball, though, the acceleration due to this drag for the orange ball is about 55 times greater than for the red ball. Thus, for the red bob, the damping is minimal, but for the orange ball it is significant. In both cases, it is linearly related to the velocity of the bob, and introduces a third term into the equation above, which now becomes

m(dx²/dt²) + c(dx/dt) + (mg/l)x = 0

The characteristic equation that corresponds to this is mr² + cr + k = 0. The roots to this equation are [-c ±√(c² - 4mk)]/2m. If we divide the equation by m, substitute 2b for c/m, and substitute ω² for g/l, the equation becomes

(dx²/dt²) + 2b(dx/dt) + ω²x = 0,

and the roots become r₁ = -b + √(b² - ω²) and r₂ = -b - √(b² - ω²). This gives rise to three different situations, depending on the relative sizes of b and ω:

1) If b² - ω² > 0, the system is said to be overdamped (or overcritically damped). The roots are real and unequal, and both are negative. The solution of the differential equation above is:

x = C₁e^r1t + C₂e^r2t.

Since both r₁ and r₂ are negative, x approaches zero as time increases. Depending on the conditions, the mass may cross the equilibrium position once, but no more than once, and the system does not oscillate. Demonstration 40.33 -- Mass-springs damped in oil, water, shows an overdamped system (the mass-spring damped in oil).

2) If b² - ω² = 0, the system is said to be critically damped. The roots are equal, and the solution of the differential equation is:

x = (C₁t + C₂)e^-ωt (or x = (C₁t + C₂)e^-bt).

As time increases, x approaches zero (but does not cross it). (The mass also approaches the equilibium position more quickly than it does when the system is overdamped.) Demonstration 40.33 -- Mass-springs damped in oil, water, shows a critically-damped system (the mass-spring damped in water).

3) If b² - ω² < 0, the system is said to be underdamped. In this case, the roots are complex and unequal. If we let ω² - b² = α², the roots are r₁ = -b + αi and r₂ = -b - αi. This leads to the solution

x = e^-bt(C₁ cos αt + C₂ sin αt).

If we set C₁ = A cos φ and C₂ = A sin φ (where A = √(C₁² + C₂²) and tan φ = C₂/C₁), we can use the trigonometric identity

A cos(αt - φ) = A cos φ cos αt + A sin φ sin αt

to obtain

x = Ae^-bt cos (αt - φ)

This equation describes damped oscillation. This is similar to simple harmonic motion whose frequency, in radians per second, is α, and whose period is T = 2π/α, except that the amplitude is not constant. Because of the factor e^-bt, which is called the damping factor, the amplitude (which equals Ae^-bt), decays over time. As noted above, b, in the damping factor, equals c/(2m). The greater the damping force (friction), the larger b is, and the more quickly the oscillations decay. We can see from the equation above, that the displacement of the mass oscillates between the two curves x = Ae^-bt and x = -Ae^-bt.

The period of oscillation, T = 2π/α = 2π/√(ω² - b²), is longer than that for the system in which there is no friction, for which T = 2π/ω. We can see that as friction, and thus b, becomes smaller, T approaches that for the frictionless system. Also, the damping factor approaches unity, and the equation above reduces to x = A cos (ωt - φ), the equation for the undamped mass-spring oscillator, given in the first paragraph above.

Both pendulums in this demonstration are underdamped, but as noted above, the damping for them is significantly different. The damping for the pendulum with the orange ball is relatively large, and the amplitude of oscillation decays noticeably within a few cycles. For the pendulum with the red bob, the damping is minimal, and it takes some time for the oscillations to decay.

References:

1) Thomas, George B., Jr. and Finney, Ross L. Calculus and Analytic Geometry (Reading, Massachusetts: Addison-Wesley Publishing Company, 1992), pp. 1081-1082.
2) Resnick, Robert and Halliday, David. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), pp. 310-311.
3) Sears, Francis Weston and Zemansky, Mark W. College Physics, Third Edition (Reading, Massachusetts: Addison-Wesley Publishing Company, 1960) p. 228-230.
4) http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1 and links to related topics.
5) You can see a simulation that shows the behavior of a pendulum, which allows you to add friction, with velocity and acceleration vectors, if you wish to display them, here: https://phet.colorado.edu/sims/html/pendulum-lab/latest/pendulum-lab_en.html. (Click on the tile named “Lab.”)