Sat May 12, 2012 By: Shivam Gupta

derive the formula for motion of a single pendulum

Expert Reply
Sun May 13, 2012

Rotational Method

pendulum variables
The pendulum is modeled as a point mass at the end of a massless rod. We define the following variables:
  • ? = angle of pendulum (0=vertical)
  • R = length of rod
  • T = tension in rod
  • m = mass of pendulum
  • g = gravitational constant
We will derive the equation of motion for the pendulum using the rotational analog of Newton's second law for motion about a fixed axis, which is ? = I ? where
  • ? = net torque
  • I = rotational inertia
  • ? = ?''= angular acceleration
The rotational inertia about the pivot is I = m R2. Torque can be calculated as the vector cross product of the position vector and the force. The magnitude of the torque due to gravity works out to be ? = ?R m g sin ?. So we have?R m g sin ? = m R2 ?which simplifies to
?'' = ? g?R sin ? (1)
This is the equation of motion for the pendulum.
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