Name: Understanding Gravitational Field
Uploaded: 2019-07-19
Duration: 5 min
Description: Each mass exerts a force on the other mass. This phenomenon is explained using the gravitational field. Let's watch this video to understand the gravitationa...

Question

Find the gravitational potential

(V)

and intensity

(E)

at a point at a distance

r

from the centre of a uniform ring of radius

a

.

Accepted Answer

Consider a ring of uniform mass distribution. The figure is given below.

The gravitational field intensity due to elemental mass $d m$ at a distance $r$ from the centre is given by, $E = \frac{G d m}{r^{2}}$

In this figure, the gravitational intensity due to the elemental mass at $P$ will be, $d E = \frac{G d m}{y^{2}}$

Where $y$ is the distance from the elemental mass to the point of consideration, $y = \sqrt{a^{2} + r^{2}}$ , Integrating to get the contribution of the whole ring,

While integrating to get the contribution of the full ring, every two diametrically opposite elemental masses, effectively create a field that is axially inward to the ring (From $P$ to $O$ ). So collecting only the axial component, (That is only the cosine of the total intensity),

$E = \int \frac{G d m}{y^{2}} \cos (θ) \Rightarrow E = \int \frac{G d m}{(a^{2} + r^{2})} \cos (θ)$

From the figure, $\cos (θ) = \frac{r}{\sqrt{a^{2} + r^{2}}}$ .

Integrating to get the total contribution, $E = \int \frac{G r}{{(a^{2} + r^{2})}^{\frac{3}{2}}} d m \Rightarrow \overset{⇀}{E} = - \frac{G m r}{{(a^{2} + r^{2})}^{\frac{3}{2}}} \overset{\land}{r}$

Gravitational intensity, $E = \frac{G m r}{{(a^{2} + r^{2})}^{\frac{3}{2}}}$ directed towards the centre of the ring.

Using the expression for gravitational intensity, the potential can be derived as,
$V (r) = - \int_{\infty}^{r} E (r) . d l = - \int_{\infty}^{r} \frac{G m r}{{(a^{2} + r^{2})}^{\frac{3}{2}}} d r \Rightarrow V = - \frac{G m}{{(a^{2} + r^{2})}^{\frac{1}{2}}}$

Find the gravitational potential $(V)$ and intensity $(E)$ at a point at a distance $r$ from the centre of a uniform ring of radius $a$ .

Important Questions on Gravitation

Learn and Prepare for any exam you want !

Find the gravitational potential V and intensity E at a point at a distance r from the centre of a uniform ring of radius a.

Important Questions on Gravitation

Learn and Prepare for any exam you want !

Find the gravitational potential $(V)$ and intensity $(E)$ at a point at a distance $r$ from the centre of a uniform ring of radius $a$ .