# Gravity due a hemispherical planet

• Yalanhar
In summary, the conversation discusses the calculation of the vertical component of gravity using spherical coordinates. The correct formula is derived by correcting a mistake in the original equation, where the denominator should be ##r^2## instead of ##R^2##.
Yalanhar
Homework Statement
find the acceleration due to gravity at the centre of a solid hemisphere.
Relevant Equations
##dg=\frac {GMdm}{r^2}##
Can someone help me? I am not sure where is my mistake

##g=\pi G\rho R##

my calculus
By symmetry, I only need to add the vertical component of gravity
##dg = \frac {GMdm}{d^2}\cdot cos\theta## (1) Where d=R
##\rho =\frac{dm}{dV} ##
##dm = \rho \cdot dV## (2)

(2) in (1)

##dg = \frac {G \rho cos\theta dV}{R^2}##

for spherical coordinate $$dV = r^2drsin\theta d\theta d\phi$$

so:
##dg=\frac {G\rho }{R^2} r^2dr sin\theta cos\theta d\theta d\phi##

##g=\frac {G\rho}{R^2} \int_0^R r^2 \, dr \int_0^\frac {\pi}{2} sin\theta cos\theta \, d\theta \int_0^{2\pi} \, d\phi##

##g = \frac {G\rho}{R^2} \cdot \frac{R^3}{3} \cdot \frac {1}{2} \cdot 2\pi##

##g = \frac {G\rho R\pi}{3}##

##g=\frac {\pi G\rho R}{3}##

Yalanhar said:
Can someone help me? I am not sure where is my mistake

Yalanhar said:
##dg = \frac {G \rho cos\theta dV}{R^2}##
Not all elements ##dV## are at distance ##R## from the origin. The denominator should be ##r^2##.

Yalanhar
kuruman said:

Not all elements ##dV## are at distance ##R## from the origin. The denominator should be ##r^2##.
oh yes. I got it now. Thanks!

## 1. How does gravity differ on a hemispherical planet compared to a spherical planet?

On a hemispherical planet, gravity is not evenly distributed like on a spherical planet. The gravitational force is strongest at the center of the hemisphere and decreases towards the edges.

## 2. Can objects roll uphill on a hemispherical planet due to the uneven distribution of gravity?

No, objects will still roll towards the center of the hemisphere due to the gravitational force. The uneven distribution of gravity does not affect the direction of motion, only the strength of the force.

## 3. How does the shape of a hemispherical planet affect the orbits of objects around it?

The orbits of objects around a hemispherical planet will be affected by the uneven distribution of gravity. Objects closer to the center of the hemisphere will have a faster orbital speed than those further away.

## 4. Is the gravitational pull on a hemispherical planet stronger or weaker than on a spherical planet?

The gravitational pull on a hemispherical planet will be stronger at the center and weaker towards the edges, compared to a spherical planet where the pull is consistent throughout.

## 5. How does the shape of a hemispherical planet affect the weight of objects on its surface?

The weight of objects on a hemispherical planet will be different at different points on the surface due to the uneven distribution of gravity. Objects will weigh more at the center of the hemisphere and less towards the edges.

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