- #1
Kaguro
- 221
- 57
- Homework Statement
- A particle moves on a plane with rθ = constant. It's transverse acceleration is zero. Force acting on it is related to r as:
(a)1/r
(b)1/r^2
(c) 1/r^3
(d) 1/r^4
- Relevant Equations
- ## \dot {\hat {r} }=\dot θ \hat {\theta}##
## \dot {\hat \theta }=-\dot {\theta} \hat r##
##\vec r=r \hat r##
##\vec v=\dot r \hat r + r \dot \theta \hat \theta##
##\vec a = (\ddot r - r \dot \theta^2)\hat r + (2 \dot r \dot \theta + r \ddot \theta)\hat \theta##
Given that,
##2 \dot r \dot \theta + r \ddot \theta =0##
Also,
##r \theta=constant##
##\Rightarrow \dot r \theta + r \dot \theta=0##
##\Rightarrow \dot r = -\frac{r \dot \theta} {\theta}##
##\Rightarrow \frac{-2r \dot \theta^2}{\theta} + r \ddot \theta=0##
##\Rightarrow \frac{2 \dot \theta^2}{\theta} = \ddot \theta##
##\Rightarrow 2 \dot \theta^2=\theta \ddot \theta##
##\Rightarrow \vec a=(\ddot r-r \dot\theta^2) \hat r##
##\Rightarrow \vec a=(\ddot r-\frac{r \theta\ddot \theta}{2}) \hat r##
##\Rightarrow \vec a=(\ddot r-\frac{(const.)\ddot \theta}{2}) \hat r##
Which is independent of r.
What is wrong with my work?
##\vec v=\dot r \hat r + r \dot \theta \hat \theta##
##\vec a = (\ddot r - r \dot \theta^2)\hat r + (2 \dot r \dot \theta + r \ddot \theta)\hat \theta##
Given that,
##2 \dot r \dot \theta + r \ddot \theta =0##
Also,
##r \theta=constant##
##\Rightarrow \dot r \theta + r \dot \theta=0##
##\Rightarrow \dot r = -\frac{r \dot \theta} {\theta}##
##\Rightarrow \frac{-2r \dot \theta^2}{\theta} + r \ddot \theta=0##
##\Rightarrow \frac{2 \dot \theta^2}{\theta} = \ddot \theta##
##\Rightarrow 2 \dot \theta^2=\theta \ddot \theta##
##\Rightarrow \vec a=(\ddot r-r \dot\theta^2) \hat r##
##\Rightarrow \vec a=(\ddot r-\frac{r \theta\ddot \theta}{2}) \hat r##
##\Rightarrow \vec a=(\ddot r-\frac{(const.)\ddot \theta}{2}) \hat r##
Which is independent of r.
What is wrong with my work?