In: Physics
A pendulum is constructed from a thin, rigid, and uniform rod with a small sphere attached to the end opposite the pivot. This arrangement is a good approximation to a simple pendulum (period = 5.87 s), because the mass of the sphere (lead) is much greater than the mass of the rod (aluminum). When the sphere is removed, the pendulum is no longer a simple pendulum, but is then a physical pendulum. What is the period of the physical pendulum?
The time period of the simple pendulum T = 3.46 s
The time period of the pendulum
T = 2π/ω ------(1)
and the angular frequency of physical pendulam
ω =√mgL/I ------(2)
The moment of inertia of thin uniform rod length L is
I = 1/3 m L2 ----- (3)
substitue eq 2 in eq 1
T = 2π√I/mgL------(4)
The center of gravity of the uniform rodlies, like uniform rod is at
L = 1/2 L ----(5)
eq 3 and eq5 in eq 4
T = √2L/3g
at this length we must deal unknown length ofD rod
period of simple pendulam given by eq i and eq2
ωsimple = 2π/Tsimple
or √g/L
Tsimple = 2π√L/g
solving this expression for D/gand substitue the result intoeq 3
By plugging the values we get
which gives us