Hi please explaint it step by steps.
It can be shown that the exact ode for the pendulum problem is d2 Ɵ/dt2+ g/Lsin Ɵ = 0.
By leeting d2 Ɵ/dt2= wdw/d Ɵ, where w is the angular velocity, solvethe ode to obtain w in terms of Ɵ .
L = 0.5m , `Theta` = π/3, g = 9.81ms-2 and compare your result with question 5 using the maples evalf command to evaluate the integral.
You need to replace `(d^2 Theta)/(dt^2)` by `(wdw)/(dTheta) ` in given ode such that:
`(wdw)/(dTheta) + (g/L)*sin Theta = 0`
You need to move `(g/L)*sin Theta` to the right side such that:
`(wdw)/(dTheta)= -(g/L)*sin Theta`
`` You need to separate the variables such that:
`wdw = -(g/L)*sin Theta dTheta`
You need to integrate both sides such that:
`int wdw = int -(g/L)*sin Theta dTheta`
You need to factor out the constant term `-(g/L)` such that:
`w^2/2 = -(g/L)*(-cos Theta) + c`
`` `w^2 = 2(g/L)*(cos Theta) + c`
`w = +- sqrt(2(g/L)*(cos Theta))+c`
You need to substitute g by `9.8m/(s^2), ` L by`0.5 m` and `Theta` by `pi/3` , such that:
`w = +-sqrt(2*(9.8/0.5)*cos (pi/3))`
`w = +-sqrt (2*19.6*(1/2)) =gt w~~+-4.427 (rad)/s`
Hence, evaluating the angular velocity for the given values g,L,`Theta` , yields `w~~+-4.427 (rad)/s` .