Find time taken by a pendulum to swing by 90degrees

[vikvaryas]

Homework Statement

A small ball is attached to a massless rod at one end. The other end is hinged such that the rod can swing freely in the vertical plane.
Find the time taken by this system to rotate from horizontal position to vertical position.
Length of rod = L
Acceleration due to gravity = g
mass of ball = m
All surfaces are frictionless.

Homework Equations

The Attempt at a Solution

Tangential acceleration of ball is gcosθ where θ is angle the rod makes with horizontal
Tangential acceleration is equal to Lα where α is rate of change of angular speed
So i have Lα=gcosθ
I am unable to proceed further. what do i do?

 

[Sunil Simha]

Firstly write the expression for angular acceleration and then try to find angular velocity by integration.

 

[vikvaryas]

Firstly write the expression for angular acceleration and then try to find angular velocity by integration.

I have obtained a differential equation d2θ/dt2 = (g/L)cosθ
How do i solve this?

 

[Sunil Simha]

I have obtained a differential equation d2θ/dt2 = (g/L)cosθ
How do i solve this?

Write it as dω/dt. But dω/dt can also be written as (dω/dθ)*(dθ/dt) or ω*dω/dθ. So now integrate and you'll obtain ω in terms of θ.

Once you get that, ω can be written as dθ/dt and if you integrate the expression relating ω and θ, you can relate θ and t.

 

[haruspex]

Write it as dω/dt. But dω/dt can also be written as (dω/dθ)*(dθ/dt) or ω*dω/dθ. So now integrate and you'll obtain ω in terms of θ.

Once you get that, ω can be written as dθ/dt and if you integrate the expression relating ω and θ, you can relate θ and t.

Easily said, but you end up with a very nasty integral. See http://en.wikipedia.org/wiki/Pendulum_(mathematics)#Arbitrary-amplitude_period . One might hope that there is a closed form solution for the specific case of a 90 degree amplitude, but I'm not aware of such.