Oscillations with an arbitrary amplitude

We have studied small oscillations of the mathematical pendulum. Next, we solve the same problem for larger values of displacements, and with this purpose in mind consider both equations of the set (3.22). Multiplying them by unit vectors i and k, respectively, and adding we obtain one equation of motion 

Of course, this equation directly follows from Fig. 3.3. Multiplying Equation (3.34) 

Taking into account the fact that ds — IdO, we have 

If T is the oscillation time of pendulum or the time the pendulum needs to swing from the angle to (half period ), then integration of Equation (3.38) gives 

As follows from Equation (3.40), the angle ij/ changes from 0 to n/l, when d varies from 0 to 00, and, therefore 

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