Question

In: Physics

A pendulum is constructed from a thin, rigid, and uniform rod with a small sphere attached...

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?

Solutions

Expert Solution

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


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