# Differential equation for a pendulum

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1. Apr 12, 2017

### Faiq

1. The problem statement, all variables and given/known data
A simple pendulum is formed by a light string of length $l$ and with a small bob $B$ of mass $m$ at one end. The strings hang from a fixed point at another end. The string makes an angle $\theta$ with the vertical at time $t$. Write down an equation of motion of $B$ perpendicular to $OB$ and state the approximation that enables you to treat this equation as an SHM equation. State the general solution of this equation.

3. The attempt at a solution
Equation of motion : $\rm \large -mgl\sin(\pi -\theta) = ml^2 \ddot \theta$
Rearranges to: $\rm \large \ddot \theta =-\frac{g}{l}sin(\pi-\theta)$
Approximation: $\rm \large sin \theta \approx \theta$ provided $\theta$ is small.
General solution: $\rm \large \theta = \theta_0sin(\sqrt{\frac{g}{l}}t)$

The reason I am confused is because I think I have misunderstood what they mean by "equation of motion of $B$ perpendicular to $OB$" and may have got all of my answers wrong.
And secondly I am well aware of finding a general solution of a differential equation of the form $\rm \small \ddot y =ay$, however, I have never seen a general solution of a differential equation of the form $\rm \small \ddot y =a(\lambda-y)$ where $\lambda$ is a constant.

Last edited: Apr 12, 2017
2. Apr 12, 2017

### Dr.D

It is not entirely clear what you are calling theta (and thus why you have involved a pi - theta argument for the sine). A figure would really help your question, along with labels on the figure.

As to the solution for the differential equation, did you really mean ddy = a*y, where a is presumed to be positive? Or did you intend to have it be negative?

The additional term a*lambda simply adds a constant to the particular solution.

3. Apr 12, 2017

### Staff: Mentor

Why are you interested in the solution to that differential equation?

4. Apr 12, 2017

### Faiq

I hope this will work.

Yes, can you show me how to add the constant to the particular solution? Like for example if the solution to the differential equation $\ddot y = ay$ where a is a constant is $y = y_0 \sin \omega t$, how will the constant term be added in the solution if the equation is changed to $\ddot y = a(\lambda-y)$? (If changes vary from equation to equation, please refer to the SHM equation given in my first post) If you find my example vague, feel free to make one of yours.
I am not interested in that particular equation. I am just concerned whether changes will occur in the solution if the concerned variable was increased or decreased by a constant term.

5. Apr 12, 2017

### Dr.D

Start by replacing sin(pi-theta) with just sin(theta); they are exactly equal.

Then, with a little bit of re-arrangement, your equation should look like
dd(theta) + w^2*sin(theta) = 0

Then make the approximation that for small theta, sin(theta) = theta, so you have
dd(theta) + w^2*theta = 0

That should be the only equation you need to solve for this problem.
I really don't know where the question about a constant term on the right came from.

6. Apr 13, 2017

### Faiq

Okay thank you that explains a lot.