# mass m is fixed toPendulum A simple pendulum consisting of a point1.the end of a massless rod (length /), whose other end is pivoted from the ceiling to letit swing freely in a vertical plane,specified by its angle o from the equilibrium position.(a) Prove that the pendulum's potential energy (measured from the equilibrium level) isas shown below. The pendulum's position can beU ()mgl (1 - cos )(b) Write down the total energy E as a function of and o. Show that by differentiatingyour expression for E with respect to t you can get the equation of motion for ¢ andthat the equation of motion is just the familiar Tmoment of inertia, and a is the angular acceleration ).(c) Assuming that the angle remains small throughout the motion, show that the motionis periodic with periodIa (where T is the torque, I is the- 2т V1/9.Tоm1 (d)To get the pendulum's period good for large oscillations as well as small,the following method: Use the above PE, findwe can usedoas a function of d. Next use11dtin the form tdo/o, to write the time for the pendulum to travel from ø = 0 to itsmaximum value (the amplitude) . Now show thatdu-Фdo2То -T TO1VI-u2I-A2u2sin (d/2) sin2 (ø/2)0is the period for small oscillation given above and A = sin(P/2). These inte-where Tograls cannot be evaluated in terms of elementary functions. However, the second integralis a standard integral called the complete elliptic integral of the first kind, sometimes de-noted K(A2), whose values are tabulated and are known to computer software such asMathematica, which calls it EllipticK (A2)].

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can you do part d please

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Step 1

Write the expression for potential energy of the system.

Step 2

Write the expression for total...

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