Question

Please answer this question for physics. 10 points

Answer 1

The problem you've presented involves calculating the gravitational force between two bodies, which in this case are Saturn and Io, one of its moons. This can be done using Newton's law of universal gravitation, which is given by the formula:

`\[ F = G (m_1 m_2)/(r^2) \]`

Where:

-
`\( F \)` is the magnitude of the gravitational force between the two masses,

- g i s the gravitational constant
`(\(6.674 * 10^(-11) \, \text{N} \cdot \text{m}^2/\text{kg}^2\))`,

-
`\( m_1 \)` and
`\( m_2 \)` are the masses of the two objects,

- r is the distance between the centers of mass of the two objects.

In vector form, the force exerted by Saturn on Io will be directed along the line connecting their centers of mass. If we assume Io is at a position
`\( \vec{r} \)`relative to Saturn, then the force vector can be given by:

`\[ \vec{F} = -G \frac{m_{\text{Saturn}} m_{\text{Io}}}{|\vec{r}|^2} \hat{r} \]`

Where \( \hat{r} \) is the unit vector in the direction from Saturn to Io. The negative sign indicates that the force is attractive and directed towards Saturn.

Given the values from your question:

`- \( m_{\text{Saturn}} = 3.86 * 10^(25) \, \text{kg} \)`

`- \( m_{\text{Io}} = 5.13 * 10^(21) \, \text{kg} \)`

-
`\( r = 4.91 * 10^9 \, \text{m} \)`

Let's calculate the magnitude of the force and then express it as a vector. To express it as a vector, I will assume a coordinate system where the position vector
`\( \vec{r} \)` points from Saturn to Io. Since the problem doesn't provide directional information, I'll use the one-dimensional case along a straight line for simplicity. If you need a specific orientation in a three-dimensional space, please provide the additional details. Now, let's proceed with the calculation.

The magnitude of the gravitational force exerted by Saturn on Io is approximately
`\( 5.48 * 10^(17) \)` Newtons. Expressed as a vector in a one-dimensional space directed towards Saturn, this force would be
`\( -5.48 * 10^(17) \)` Newtons, where the negative sign indicates that the force is attractive and pointing towards Saturn. If you need a vector representation in a two or three-dimensional coordinate system, please provide the direction or orientation details.