The period of a simple pendulum, whose bob is a hollow metallic sphere, is T. The period is T1 when the bob is filled with sand, T2 when it is filled with mercury and T3 when it is half filled with mercury. Which of the following is true:
In Hollow Sphere (T) , Solid Spheres ( T1 and T2) Center of mass lie at center. Where as in Half filled sphere center of mass lie below center(T3).
As length of pendulum will increase in T3. Time period is highest for it.
Two simple pendulums of length 1 m and 16m respectively are both given small displacement in the same direction at the same instant. They will be again in phase after the shorter pendulum has completed n oscillations. The value of n is :-
\[ T = 2\pi\sqrt{\frac{l}{g}} \]
\[ T_{1}= 2\pi\sqrt{\frac{1}{g}}= 2 sec \]
\[ T_{2}= 2\pi\sqrt{\frac{16}{g}}= 8 sec \]
A simple pendulum 50 cm long is suspended from the roof of a cart accelerating in the horizontal direction with constant acceleration √3g m / s² .The period of small oscillations of the pendulum about its equilibrium position is (g = π² m/s²):
\[ T = 2\pi\sqrt{\frac{l}{\sqrt{a^{2}+g^{2}}}} \]
\[ T = 2\pi\sqrt{\frac{0.5}{\sqrt{\left( \sqrt{3}g \right)^{2}+g^{2}}}} \]
\[ T = 2\pi\sqrt{\frac{0.5}{2g}} \]
\[ T = 2\pi\frac{1}{2\sqrt{g}} = 2\pi\frac{1}{2\pi}= 1 sec \]