Question

A simple pendulum of length l has maximum angular displacement theta . Then maximum kinetic energy of a bob of mass m is

Answer 1

The maximum Kinetic energy of bob =             mg(L - Lcosθ)

Step by step explanation:

$\textbf{According to conservation of energy, The total energy of a system } \\\textbf{can neither be created nor destroyed. }$

So the Total energy (E) = Kinetic energy (KE)  + Potential energy (PE)

                                                                                              (equation 1)

At the mximum height ,the  speed of bob is 0, so KE is 0 but the PE is maximum, and

at the minmum height, the speed of bob is maximum so KE is maximum, but PE is  0.

So by equation 1,

Total energy at minimum height = KE + 0 = KE

Total energy at maximum height = 0 + PE = PE               (equation 2)

from equation 1, conservation of total energy

 Total energy at minimum height =Total energy at maximum height

KE  = PE                                                                        (equation 3)

By observing the figure,  

The height between minimum and maximun height = L - Lcosθ(equation 4)

$\textbf{\Large The formula of Potential energy =}m\times g\times h$

by equation 4,

So, PE = $m\times g\times (L - L\cos \theta)$

By equation 3

$\textbf{\Large The maximum Kinetic energy of bob =} m\times g\times (L - L\cos \theta)$