Analytic mechanics in polar coordinates

[chingcx]

Homework Statement

A mass follows the path of a cardioid r=1+sinφ with given speed, what is its period?

Homework Equations

The Attempt at a Solution

I attempt to do an integral on polar coordinates to find the distance covered by the mass first.
The integral I derived is
$$\int_0^{2\pi} \sqrt{r^{2}+(\frac{dr}{d\varphi})^2}d\varphi$$
A mistake here? Or there are other methods to find the time?
Any help is appreciated.

UPDATE: I figure out the integral, but it is zero (ok, I know it's correct ya), but certainly not something I want.

 

[siddharth]

Homework Statement

A mass follows the path of a cardioid r=1+sinφ with given speed, what is its period?

Homework Equations

The Attempt at a Solution

I attempt to do an integral on polar coordinates to find the distance covered by the mass first.
The integral I derived is
$$\int_0^{2\pi} \sqrt{r^{2}+(\frac{dr}{d\varphi})^2}d\varphi$$
A mistake here? Or there are other methods to find the time?
Any help is appreciated.

UPDATE: I figure out the integral, but it is zero (ok, I know it's correct ya), but certainly not something I want.

The infinitesimal arc length in polar coordinates $dl$ is what you integrated over in your expression. ie,

$$dl= \sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$

If you integrate this, it gives you the total arc-length. If the speed is constant, what is the time required to travel a distance dl? From this, you can get the period.

Hint: Remember that

$$\left(\sin\frac{\theta}{2} + \cos\frac{\theta}{2}\right)^2 = 1+\sin{\theta}$$