Question

A satellite has a mass of 372 kg and is located at 2.01 ✕ 106 m above the surface of Earth. (a) What is the potential energy associated with the satellite at this location? Your response differs from the correct answer by more than 10%. Double check your calculations. J (b) What is the magnitude of the gravitational force on the satellite?

Answer 1

The potential energy of the satellite is approximately 7.17 x 10^9 J, and the magnitude of the gravitational force on the satellite is 3,645.6 N.

Step-by-step explanation:

To calculate the potential energy of the satellite, we can use the formula:

Potential Energy = mass * gravitational field strength * height

Given that the mass of the satellite is 372 kg, the height is 2.01 x 10^6 m, and the gravitational field strength is 9.8 m/s^2, we can substitute these values into the formula to calculate the potential energy:

Potential Energy = 372 kg * 9.8 m/s^2 * 2.01 x 10^6 m

This calculation results in a potential energy of approximately 7,174,416,800 J, which is 7.17 x 10^9 J.

To find the magnitude of the gravitational force on the satellite, we can use the formula:

Gravitational Force = mass * gravitational field strength

Substituting the given values into the formula, we get:

Gravitational Force = 372 kg * 9.8 m/s^2

Calculating this yields a gravitational force of 3,645.6 N.

Answer 2

Gravitational potential energy of the satellite: approximately
`(\rm -1.77 * 10^(10)\; J)`. (Note that this value is negative.)

Magnitude of the gravitational force between the earth and the satellite: approximately
`\rm 2.11* 10^(3)\; N`

Step-by-step explanation:

(a)

Consider the formula for the gravitational potential energy (GPE) between two dot or spherical mass:

`\displaystyle \text{GPE} = -(G \cdot m \cdot M)/(r)`,

where

• 
  `G \approx \rm 6.67 * 10^(-11)\; N \cdot m^(2) \cdot kg^(-2)` is the gravitational constant,
• 
  `m` and
  `M` are the mass of the two objects, and
• 
  `r` is the separation between the center of mass of the two objects.

Note the negative sign in front of the fraction. Electrostatic force might either be attractive or repulsive. However, gravity between two object of mass is always attractive. As a result, unlike electrical potential energy, the gravitational potential energy between two objects with mass should always be negative.

Keep in mind that the center of mass of a sphere with uniform density is at its center. If the earth is approximated as a sphere like that, its center of mass wouldn't be on the surface (i.e. the ground.) Rather, the center of the earth's mass should be somewhere deep underground near its core.

Look up the following quantities:

• Mean radius of the earth (average distance between the core and the surface:) approximately
  `\rm 6.371 * 10^(6)\; m`.
• Mass of the earth: approximately
  `\rm 5.972 * 10^(24)\; kg`.

The distance between the satellite and the center of mass of the earth comes in two parts:

• distance between the satellite and the surface of the earth, and
• distance between the surface of the earth and its center of mass.

In other words,

`r = \rm \underbrace{\rm 2.01 * 10^(6)\; m}_{\text{Satellite and ground}} + \underbrace{\rm 5.972 * 10^(6)\; m}_{\text{Ground and Center of Mass}} \approx 7.98 * 10^(6)\; m`.

Make sure that all values are in standard units and apply the formula:

`\begin{aligned}\text{GPE} &= -(G \cdot m \cdot M)/(r) \\ &= - (6.67 * 10^(-11)* 372 * 5.972 * 10^(24))/(6.371 * 10^(6)) \\ & \rm \approx -1.77 * 10^(10)\; J\end{aligned}`.

Since all inputs are in their standard units, the unit of the output (potential energy) should be in joules, which is the standard unit of energy.

(b)

The formula for the magnitude of gravitational force between two points (or sphere) of mass is

`\displaystyle \text{Gravitational Force} = (G \cdot m \cdot M)/(r)^(2)`.

Note that as a magnitude, the output of this formula should always be positive. Evaluate this formula for the same
`G`,
`m`,
`M`, and
`r` as in part (a) of this problem:

`\begin{aligned}\text{Gravitational Force}&= (G \cdot m \cdot M)/(r^(2)) \\ &= (6.67 * 10^(-11)* 372 * 5.972 * 10^(24))/(\left(6.371 * 10^(6)\right)^(2)) \\ & \rm \approx 2.11 * 10^(3)\; N\end{aligned}`.