Suspended from one rod are pendulums of equal mass but different lengths, and from the other rod, pendulums of equal length but unequal mass. When you set the two pendulums of different length oscillating, they swing at different frequencies. When you set the two pendulums of the same length oscillating, they swing at the same frequency.

The pendulums in this demonstration are essentially point masses at the ends of strings, that is, simple pendulums. The two red pendulum bobs have a mass of 538 g, and the distances from the suspension points to their centers of mass are 1.0 meters and 0.50 meters. The mass of the brass bob is 239 g, and the mass of the bowling ball is 7.3 kg. For both of these, the distance from suspension point to center of mass is 0.96 meters. (For the bowling ball there is an added subtlety, in that its center of mass is a significant distance from the point at which it is attached to the rope. The bowling ball thus rotates about that point as it swings. This rotation introduces a small deviation of its behavior from that of a simple pendulum.) The force on the pendulum bob is that of gravity, F = mg, which is balanced by the tension in the string or rope 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). 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.

One can show that for large angles of displacement, the period of the pendulum is given by the following equation, in which θ_max is the maximum angular displacement of the pendulum:

T = 2π√(l/g)[1 + (1/2²) · sin²(θ_max/2) + (1/2²) · (3²/4²) · sin⁴(θ_max/2) + · · ·].

Galileo was the first to note the dependence of the period of a pendulum on its length. As Dava Sobel notes in Longitude,

Legends of Galileo recount an early mystical experience in church that fostered his profound insights about the pendulum as a timekeeper: Mesmerized by the to-and-fro of an oil lamp suspended from the nave ceiling and pushed by drafts, he watched as the sexton stopped the pan to light the wick. Rekindled and released with a shove, the chandelier began to swing again, describing a larger arc this time. Timing the motion of the lamp with his own pulse, Galileo saw that the length of a pendulum determines its rate.

As Sobel further notes, Galileo had intended to use the pendulum as the basis for a clock. His son, Vincenzio, made a model from his drawings, and the city fathers of Florence later built a tower clock based on that design. Christiaan Huygens, however, was the first actually to make a working pendulum clock.

When you displace the pendulum bob from vertical by an angle θ, through a distance x, you raise it to a height h₀ = x tan θ. It now has potential energy U = mgh₀, and until you release it, its kinetic energy is K (= (1/2)mv²) = 0. When you release the bob, gravity accelerates it toward its equilibrium position (θ = 0). At this point, h = 0, and all of the original potential energy has been converted into kinetic energy. K = (1/2)mv² = mgh₀, and v = √(2gh₀). The potential energy is zero, and the kinetic energy is at its maximum (as is v). As the bob continues through its swing, the kinetic energy is converted back to potential energy, until the bob reaches its original height at the other end. At this point, the kinetic energy is again zero, and the potential energy is again mgh. Over the entire swing of the pendulum bob, the sum of its potential energy and kinetic energy is constant.

It is also interesting to note the relationships among position, velocity and acceleration of the pendulum bob. Strictly speaking, at the release point the acceleration vector is tangent to the path of the pendulum bob, pointing toward the middle of the swing. After the bob is released, this vector rotates until it is vertical (pointing upward) at the bottom of the swing, and again tangent to the path at the opposite turning point, pointing back toward the middle of the swing. (The acceleration is due to the combined effects on the bob of gravity and the tension in the string from which it is suspended.) The velocity vector is always tangent to the path of the pendulum bob (and always points in the direction in which the bob is moving). With the approximation that we made above for small θ, however, we may treat the motion of the bob as being nearly straight-line motion along the x-axis, and consider only the x-component of the acceleration. If we take x = x₀ cos (ωt - φ) (where x₀ = lθ_max), then v = dx/dt = -ωx₀ sin (ωt - φ), and a = d²x/dt² = -ω²x₀ cos (ωt - φ). We see that when the bob is at the turning points (maximum angular displacement to either side), the velocity is zero and the acceleration (in the x direction) is maximum, and at the bottom of the swing (where the pendulum is vertical), the acceleration (in the x direction) is zero and the velocity is maximum.

References:

1) Resnick, Robert and Halliday, David. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), pp. 310-311.
2) Sears, Francis Weston and Zemansky, Mark W. College Physics, Third Edition (Reading, Massachusetts: Addison-Wesley Publishing Company, 1960) p. 228-230.
3) http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1 and links to related topics.
4) Sobel, Dava Longitude (New York: Walker Publishing Company, 1995), pp. 36-37.
5) You can see a simulation that shows the behavior of a pendulum, 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.”)