Suppose a pendulum with length L (meters) has angle theta (radians) from the ver

Question

Suppose a pendulum with length L (meters) has angle theta (radians) from the vertical. It can be shown that theta as a function of time satisfies the differential equation: d^2 theta/dt^2 + 8/L sin theta = 0 where g = 9.8 m/sec/sec is the acceleration due to gravity. For small values of theta we can use the approximation sin(theta) ~ theta, and with that substitution, the differential equation becomes linear. A. Determine the equation of motion of a pendulum with length 1 meters and initial angle 0.3 radians and initial angular velocity d theta/dt 0.5 radians/sec. B. At what time does the pendulum first reach its maximum angle from vertical? (You may want to use an inverse trig function in your answer) seconds C. What is the maximum angle (in radians) from vertical? D. How long after reaching its maximum angle until the pendulum reaches maximum deflection in the other direction? seconds E. What is the period of the pendulum, that is the time for one swing back and forth? seconds

Explanation / Answer

Here its characterstic equation will be given as :

r^2+ 9.8/1=0

or r=+-root(9.8)

So its solution will be

theta = c1cos(root(9.8)t)+ c2sin(root(9.8)t) ...........(i)

Now initially theta =0.3 and t=0, so clearly

0.3 = c1

on differentiating (i) again, we get

d(theta)/dt = root(9.8)(-0.3sin(root(9.8)t + c2cos(root9.8)t)

Now as d(theta)/dt ) = 0.5 and t=0, then

0.5 =root(9.8) ( -0.3 sin(0) + c2 cos(0)) i.e. c2= 0.5/root(9.8) = 0.16

So required equation is :

theta = 0.3 cos(root(9.8)t) + 0.16 sin(root(9.8)t) ...............(ii)

This is the answer of part (a).

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Now for max. angle

d(theta)/dt = root(9.8)(-0.3sin(root(9.8)t + 0.16cos(root9.8)t) =0

or sin(root(9.8)t/cos(root(9.8)t= 0.16/0.3 = 0.5333

or tan (root(9.8) t) = 0.5333

or 3.13t = tan^(-1) (0.5333) = 0.49

or t= 0.49 /3.13 = 0.156

so required time is 0.156 seconds.

This is the answer of part (b)

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and on plug in this value of t in equation (ii), we get

theta = 0.3 cos(root(9.8)(0.156) + 0.16 sin(root(9.8)(0.156)

\=0.3 cos(0.49) + 0.16 sin(0.49 ) = 0.3(0.88236) + 0.16(0.47) = 0.2647 +0.0752

=0.34 radian

that is the maximum angle.

This is the answer of part (c)

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Now in covering one period, a pendulam moves four rounds coming back to its initial position. Now as time taken in one covering max. angle is calculated as 0.156 seconds. So total time taken in one period by the pendulam

= 4 x 0.156 = 0.624 seconds.

This is the answer of part (e)

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