Direct substitution in the analysis of periodic motion

[gsmith12]

Homework Statement

By direct substitution, show that equation (3) is a solution of the differential equation (2).

Homework Equations

(2) (d^2 θ)/(dt^2 )=-g/l θ (Second derivative of θ(t)=-g/l θ.)


(3) θ(t)=θ_0 cos⁡(√(g/l) t)

The Attempt at a Solution

I tried to integrate equation (2) and derive equation (3) but it didn't come out correctly.

 

[tiny-tim]

Welcome to PF!

By direct substitution, show that equation (3) is a solution of the differential equation (2).

(2) (d^2 θ)/(dt^2 )=-g/l θ (Second derivative of θ(t)=-g/l θ.)

(3) θ(t)=θ_0 cos⁡(√(g/l) t)
…
I tried to integrate equation (2) and derive equation (3) but it didn't come out correctly.

Hi gsmith12! Welcome to PF!

i] The question doesn't want you to integrate …

it says use direct substitution … which means simply put θ(t) = θ₀cos⁡(√(g/l) t) into (d^2 θ)/(dt^2 ), and show that it comes out as -g/l θ

ii] but if you still want to integrate, multiply both sides by dθ/dt first

 

[gsmith12]

Thanks for the help. I think I am on the right track but am still running into a bit a difficulty .

To plug equation (3) into equation (2) do I first need to solve for theta? I am sorry but I am not completely clear on the set up for the direct substitution.

Thanks

 

[tiny-tim]

welcome to reality!

To plug equation (3) into equation (2) do I first need to solve for theta? I am sorry but I am not completely clear on the set up for the direct substitution.

Hi gsmith12!

I think you're slightly in denial about reality …

(3) θ(t)=θ_0 cos⁡(√(g/l) t)

is a solution …

equation (3) has solved for θ.

So just plug-and-play!