Question

A particle describes the curve r = ae^theta with constant angular velocity. Show
that its radial acceleration is zero and the transverse acceleration varies as its
distance from the pole.

​

Answer 1

SOLUTION

TO PROVE

A particle describes the curve $\displaystyle\sf{ \:r = a \: {e}^{ \theta} }$ with constant angular velocity. Show that its radial acceleration is zero and the transverse acceleration varies as it's distance from the pole.

EVALUATION

Here the given curve is

$\displaystyle\sf{ \:r = a \: {e}^{ \theta} }$

Now

$\displaystyle\sf{ Angular \: Velocity = \frac{d\theta}{dt} = \omega = constant }$

Here

$\displaystyle\sf{ \:r = a \: {e}^{ \theta} }$

$\displaystyle\sf{ \implies \: \frac{dr}{dt} = a \: {e}^{ \theta} \frac{d\theta}{dt} = r \omega}$

$\displaystyle\sf{ \therefore \: \frac{ {d}^{2} r}{d {t}^{2} } =\omega \frac{dr}{dt} =\omega. r \omega = {\omega}^{2}r}$

∴ Radial acceleration

$\sf{ = f_r}$

$\displaystyle\sf{ = \frac{ {d}^{2} r}{d {t}^{2} } - r {\bigg( \frac{dr}{dt} \bigg) }^{2} }$

$\displaystyle\sf{ = {\omega}^{2}r - r{\omega}^{2}}$

$= 0$

∴ Transverse acceleration

$\sf{ = f_{ \theta} }$

$\displaystyle\sf{ = \frac{1}{r} \frac{d}{dt} \bigg( {r}^{2} \frac{d\theta}{dt} \bigg)}$

$\displaystyle\sf{ = \frac{1}{r} \frac{d}{dt} \bigg( {r}^{2} \omega \bigg)}$

$\displaystyle\sf{ = \frac{ \omega }{r} \frac{d}{dt} \bigg( {r}^{2} \bigg)}$

$\displaystyle\sf{ = \frac{ \omega }{r} .2r \frac{dr}{dt} }$

$\displaystyle\sf{ = \frac{ \omega }{r} .2r .r \omega }$

$\displaystyle\sf{ =2 {\omega}^{2} r }$

$\displaystyle\sf{ \therefore \: \: f_{ \theta} =2 {\omega}^{2} r }$

$\displaystyle\sf{ \therefore \: \: f_{ \theta} \propto \: r }$

∴ Radial acceleration = 0

$\displaystyle\sf{ \therefore \: \: Transverse \: \: acceleration \propto \: r }$

Hence radial acceleration is zero and the transverse acceleration varies as it's distance from the pole.

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. write the expression for gradient and divergence

https://brainly.in/question/32412615

2. prove that the curl of the gradient of

(scalar function) is zero

https://brainly.in/question/19412715