Pendulum Cord Tension
Question:
A pendulum consists of a weight of mass m, called a bob, on a
cord of length L and negligible mass. Initially the pendulum is
held such that the cord is tight and horizontal. What is the
tension in the cord as a function of the angle (q) through which the pendulum has descended
after it is released.
Answer:
The velocity of the bob at any point in the swing will be such
that the change in kinetic energy 1/2*m*v2 is equal to
the change in potential energy of m*g*L*sin(q), or
v2=m*g*L*sin(q)/(1/2*m), or
v2=2*g*L*sin(q).
The centripetal acceleration at any point in the swing will be
v2/L=2*g*sin(q).
The acceleration due to gravity is g, so the component
contributing to tension in the cord is g*sin(q). The tension in the cord is the mass of
the bob times the acceleration provided by the cord, or
T=m*(2*g*sin(q)+g*sin(q))=3*m*g*sin(q).
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