. m × . m ) = . J which is same as the sum of the two energies at a displacement of cm.
This is in conformity with the principle of conservation of energy. ⊳ . The Simple Pendulum It is said that Galileo measured the periods of a swinging chandelier in a church by his pulse beats. He observed that the motion of the chandelier was periodic.
The system is a kind of pendulum. You can also make your own pendulum by tying a piece of stone to a long unstretchable thread, approximately cm long. Suspend your pendulum from a suitable support so that it is free to oscillate. Displace the stone to one side by a small distance and Fig.
. (a) A bob oscillating about its mean position. (b) The radial force T-mg cos θ provides centripetal force but no torque about the support. The tangential force mg sin θ provides the restoring torque.
the mean position. Consider simple pendulum — a small bob of mass m tied to an inextensible massless string of length L . The other end of the string is fixed to a rigid support. The bob oscillates in a plane about the vertical line through the support.
Fig. . (a) shows this system. Fig.
. (b) is a kind of ‘free-body’ diagram of the simple pendulum showing the forces acting on the bob. Let θ be the angle made by the string with the vertical. When the bob is at the mean position, θ = There are only two forces acting on the bob; the tension T along the string and the vertical force due to gravity (=mg).
The force mg can be resolved into the component mg cos θ along the string and mg sin θ perpendicular to it. Since the motion of the bob is along a circle of length L and centre at the support point, the bob has a radial acceleration ( ω L ) and also a tangental acceleration; the latter arises since motion along the arc of the circle is not uniform.