New formula for centripital force ? whats wrong

[ManishR]

new formula for centripetal force ? what's wrong !

consider a circular motion with following variables with usual meanings,
$$\vec{r},\vec{F},\overrightarrow{\theta},t,v$$

$$v=r\frac{d\theta}{dt}$$

now

$$\frac{d\hat{r}}{dt}=(\frac{d\theta}{dt})\hat{\theta}$$

$$\Rightarrow\frac{d\hat{r}}{dt}=\frac{v}{r}\hat{\theta}$$

$$\Rightarrow\frac{d^{2}\hat{r}}{dt^{2}}=-\frac{v}{r}\hat{r}$$

now according to Newton's law

$$m\frac{d^{2}\overrightarrow{r}}{dt^{2}}=\overrightarrow{F}$$

$$\Rightarrow mr\frac{d^{2}\hat{r}}{dt^{2}}=\overrightarrow{F}$$

$$\Rightarrow-mv\hat{r}=\overrightarrow{F}$$

i am still not sure what actually this equation saying.
can someone recheck it please ? where i gone wrong ?

 

[K^2]

You've made a mistake taking $\frac{d\hat{\theta}}{dt}$.

$$\frac{d\hat{\theta}}{dt} = -\left(\frac{d\theta}{dt}\right)\hat{r}$$

So

$$\frac{d^2\hat{r}}{dt^2} = - \left(\frac{v}{r}\right)^2 \hat{r}$$

And then

$$\vec{F} = mr\frac{d^2\hat{r}}{dt^2} = -\frac{mv^2}{r}\hat{r}$$

Which is the correct formula for centripetal force.

 

[ManishR]

You've made a mistake taking $\frac{d\hat{\theta}}{dt}$.

$$\frac{d\hat{\theta}}{dt} = -\left(\frac{d\theta}{dt}\right)\hat{r}$$

So

$$\frac{d^2\hat{r}}{dt^2} = - \left(\frac{v}{r}\right)^2 \hat{r}$$

And then

$$\vec{F} = mr\frac{d^2\hat{r}}{dt^2} = -\frac{mv^2}{r}\hat{r}$$

Which is the correct formula for centripetal force.

thank u so much for ur help.

 

[litup]

You've made a mistake taking $\frac{d\hat{\theta}}{dt}$.

$$\frac{d\hat{\theta}}{dt} = -\left(\frac{d\theta}{dt}\right)\hat{r}$$

So

$$\frac{d^2\hat{r}}{dt^2} = - \left(\frac{v}{r}\right)^2 \hat{r}$$

And then

$$\vec{F} = mr\frac{d^2\hat{r}}{dt^2} = -\frac{mv^2}{r}\hat{r}$$

Which is the correct formula for centripetal force.

What is the difference between the two 'r''s? r by itself and r ^?

 

[Doc Al]

What is the difference between the two 'r''s? r by itself and r ^?

r is a magnitude; $\hat{r}$ is a unit vector.