Question

The planet Jupiter is more than 300 times as massive as Earth, so it might seem that a body on the surface of Jupiter would weigh 300 times as much as on Earth. But it so happens that a body would weigh scarcely 3 times as much on the surface of Jupiter as on the surface of Earth. Discuss why this is so, using the terms in the equation for gravitational force to guide your thinking.

Answer 1

Answer:

See discussion below.

Explanation:

The weight of any body on the planet is the gravitational force of attraction between the planet and the body.

For Earth, Newton's law of gravitation expresses this force as

$F = G\dfrac{M_Em}{R_E^2}$

$M_E$ is the mass of the Earth, $m$ is the mass of the body and $R_E$ is the radius of the Earth which is the distance between the body and the Earth if it's on the Earth's surface. $G$ is the universal gravitational constant.

The weight of the body on Earth is

$W=mg_E$

Here, $g_E$ is the gravitational acceleration on Earth.

Since both forces are equal,

$mg_E = G\dfrac{M_Em}{R_E^2}$

$g_E = G\dfrac{M_E}{R_E^2}$

It is seen that $g_E$ depends only on the mass and radius of the Earth.

From the question, the mass of Jupiter is over 300 times that of the Earth. Let's name it $M_J$. Hence,

$M_J=300M_E$

If we denote Jupiter's gravitational acceleration with $g_J$ and it's radius by $R_J$, then

$g_J = G\dfrac{M_J}{R_J^2}$

$g_J = G\dfrac{300M_E}{R_J^2}$

It would follow that this is 300 times as much that of earth. However, the radius of Jupiter is also much greater than the radius of the Earth. In fact, it is about 11 times greater i.e. $R_J=11R_E$.

Hence,

$g_J = G\dfrac{300M_E}{(11R_E)^2}$

$g_J = G\dfrac{300M_E}{121R_E^2}$

$g_J = \dfrac{300}{121}G\dfrac{M_E}{R_E^2}$

$g_J = 1.5G\dfrac{M_E}{R_E^2}$

$g_J = 1.5g_E$

It is thus seen that its gravity is only about 1.5 times that if the Earth. Hence, any object on Jupiter would only weigh 1.5 times as much as it would on Earth.

Answer 2

Answer:

Yes it is true that bodies only weigh 3 times more on Jupiter as they do on Earth. I will explain below.

Explanation:

By definition, weight is a product of mass and acceleration due to gravity, i.e

$W = m * g$

As we know from physics, mass is constant. So if the body's weight increases, it should be because g has changed on the different planet. Every planet has their acceleration due to gravity.

Now, from gravitational physics, acceleration due to gravity (g) is related to the gravitational constant (G) as follows:

$g = \frac{GM}{r^{2} }$

where M is mass and r is radius of planet.

So defining the weight of the body on both planets as $W_{J}$ and $W_{E}$ for Jupiter and Earth respectively.

The problem has stated that the ratio $\frac{W_{J}}{W_{E}}$  is roughly equal to 3. to confirm this, we use the above equations as follows:

$\frac{W_{J}}{W_{E}} = \frac{mg_{j} }{mg_{E}} =\frac{ \frac{GM_{J} }{r^{2}_{J} }}{\frac{GM_{E} }{r^{2}_{E} }}$

This gives

$\frac{W_{J}}{W_{E}} =\frac{M_{J}r^{2}_{E} }{M_{E}r^{2}_{J} }}$

As you must have seen from the question, Jupiter is 300 times heavier than Earth ($\frac{M_{J} }{M_{E} }}$ =300 approximately), and by checking the radius of the Earth and Jupiter, which are respectively 6400km and 69000km roughly. So $\frac{r^{2}_{E} }{r^{2}_{J} }}$  is roughly 1/116. This means that

$\frac{W_{J}}{W_{E}}$ is approximately (300 * 1/120) = 2.59 or 3 approximately as claimed.

This means you would weigh 3 times as much on Jupiter as you weigh on Earth.