Direct substitution in the analysis of periodic motion

In summary, the given problem asks to show that equation (3) is a solution of the differential equation (2) by using direct substitution. This means that we simply plug in the given equation (3) for θ(t) in equation (2) and show that it satisfies the equation. There is no need to solve for θ as it is already given in equation (3).
  • #1
gsmith12
2
0

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.
 
Physics news on Phys.org
  • #2
Welcome to PF!

gsmith12 said:
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! :smile:

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 :wink:

ii] but if you still want to integrate, multiply both sides by dθ/dt first :smile:
 
  • #3
Thanks for the help. I think I am on the right track but am still running into a bit a difficulty :confused:.

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
 
  • #4
welcome to reality!

gsmith12 said:
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! :smile:

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. :smile:

So just plug-and-play! :wink:
 

1. What is direct substitution in the analysis of periodic motion?

Direct substitution is a mathematical method used to analyze and solve equations related to periodic motion. It involves plugging in known values for variables in an equation to find the value of the unknown variable.

2. How is direct substitution used in the study of periodic motion?

Direct substitution is used to solve equations related to periodic motion, such as equations for displacement, velocity, and acceleration. By plugging in known values for time, amplitude, frequency, or other variables, we can find the value of the unknown variable and better understand the motion of an object.

3. What are the advantages of using direct substitution in the analysis of periodic motion?

Direct substitution allows us to easily solve equations for unknown variables in periodic motion without having to use more complicated methods such as integration or differentiation. It also provides a straightforward way to analyze and compare different aspects of periodic motion, such as displacement and velocity, for a given time or set of conditions.

4. Are there any limitations to using direct substitution in the analysis of periodic motion?

Direct substitution is most effective for simple periodic motion, and may not be as useful for more complex or nonlinear motion. It also assumes that the motion is perfectly periodic, which may not always be the case in real-world situations.

5. How can direct substitution be applied to real-world scenarios?

Direct substitution can be used in a variety of real-world scenarios, such as analyzing the motion of a pendulum, a mass on a spring, or a simple harmonic oscillator. It can also be applied to more complex systems by breaking them down into smaller, periodic components and using direct substitution to analyze each component separately.

Similar threads

  • Introductory Physics Homework Help
2
Replies
38
Views
1K
  • Introductory Physics Homework Help
Replies
29
Views
1K
  • Introductory Physics Homework Help
2
Replies
51
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
14
Views
463
  • Introductory Physics Homework Help
Replies
6
Views
2K
  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
1K
  • Introductory Physics Homework Help
Replies
4
Views
2K
  • Introductory Physics Homework Help
Replies
11
Views
4K
Back
Top