A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass M = 1.61 kg and length L = 0.153 m that is pivoted freely about one end, with a solid disk of mass 3M and a radius of L4𝐿4 attached to the free end of the rod. Determine the length L that gives a period of T = 2.45 s.
Question
A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass M = 1.61 kg and length L = 0.153 m that is pivoted freely about one end, with a solid disk of mass 3M and a radius of L4𝐿4 attached to the free end of the rod. Determine the length L that gives a period of T = 2.45 s.
Solution
The period of a physical pendulum is given by the formula:
T = 2π √(I / (m * g * d))
where:
- T is the period,
- I is the moment of inertia of the pendulum,
- m is the total mass of the pendulum,
- g is the acceleration due to gravity, and
- d is the distance from the pivot point to the center of mass of the pendulum.
The moment of inertia I of the pendulum is the sum of the moment of inertia of the rod and the disk. The moment of inertia of a rod pivoted about one end is (1/3)ML² and the moment of inertia of a disk about an axis through its center is (1/2)MR². Therefore:
I = (1/3)ML² + (1/2)(3M)(L/4)²
The total mass m of the pendulum is the sum of the mass of the rod and the disk, which is M + 3M = 4M.
The distance d from the pivot point to the center of
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