How do we Determine the Mass of a Planet?

Activity I: *How do objects move in a circular orbit?*

In this experiment we will verify how objects such as planets move in a
circular orbit. The apparatus consists of a long string threaded through
a short tube, one end connected to a rubber stopper and the other end to
a hanging mass.

![](media/image1.jpeg){width="2.4722222222222223in"
height="2.0493055555555557in"}

***[Procedure:]{.underline}***

1.  Attach a hanging mass at the end of the string.

2.  Mark a spot on the string about 1 inch below the tube.

3.  Twirl the stopper above your head so that the mark does not move up
    or down.

4.  Change the speed at which you rotate the rubber stopper so that the
    mark moves up and down?

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# What happens to the mark when you speed up the stopper? \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

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# What happens to the mark when you slow down the stopper? \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

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**Why does the hanging mass not fall down?**
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*[Theory]{.underline}*

In addition to his theories on gravity, Isaac Newton gave us several
Laws that explain how objects move. Newton's first law says that an
object moving in a straight line will continue to move in a straight
line unless acted on by another force. For an object to move in a circle
it needs to change its direction, otherwise it would continue moving in
a straight line tangential to the circle. To have an object move in a
circle a force needs to be applied towards the center.

![](media/image2.jpeg){width="3.25in" height="2.954861111111111in"}

This force is called the centripetal force (*F~c~*) and it is given by
the equation

$$F_{c} = \frac{\text{mv}^{2}}{r}$$

where ***m*** is the mass of the object moving in a circle, ***v*** is
the velocity of the object, and ***r*** is the radius of the circle.

In the case of the previous activity the hanging mass provided the force
that pointed towards the center, allowing the stopper to move in a
circular orbit.

![](media/image3.png)

• If the hanging mass is not moving up or down the weight of the hanging
mass (*F~g~*) is equal to the centripetal force (*F~c~*).

• If you spin the stopper too fast the centripetal force will be greater
than the weight of the hanging mass and the hanging mass will move up.

• If you spin too slowly is smaller centripetal force and the mass
slides down.

**Activity II.**

Repeat the experiment

***[Procedure:]{.underline}***

1.  Measure the mass of the mass hanging at the end of the string.

2.  Calculate the weight of the hanging mass.

> The weight is equal to the mass times the acceleration from gravity.
>
> *F~g~ = Mg*

3.  Measure the distance between the tube and rubber stopper (***r***).

4.  Mark a spot on the string about 1 inch below the tube.

5.  Twirl the stopper above your head so that the mark does not move up
    or down.

6.  Record the time it takes for the stopper to go around once.

> It might be easier if you measure the time it takes the stopper to go
> around 10 times ***(T~10~)*** and divide by 10.

7.  Calculate the velocity.

> $$\text{velocity} = \frac{\text{dis}\text{tan}\text{ce}}{\text{time}} = \frac{2\pi r}{T}$$

8.  Calculate the centripetal force

> $$F_{c} = \frac{\text{mv}^{2}}{r}$$

9.  Repeat with a different hanging mass.

***[Data:]{.underline}***

Mass of Stopper (m): \_\_\_\_\_\_\_\_\_\_\_\_ kg

+-----+------+--------+-----+-----+-----+------+---------+---------+
| Tr  | M    | $$F    | r   | T~  | T   | $$v  | $$F_{c} | $$\fr   |
| ial |      | _{g} = |     | 10~ |     | = \f |  = \fra | ac{r}{T |
|     | (kg) |  \text | \(  |     | \(  | rac{ | c{\text | ^{2}}$$ |
|     |      | {Mg}$$ | m\) | \(  | s\) | 2\pi | {mv}^{2 |         |
|     |      |        |     | s\) |     |  r}{ | }}{r}$$ |         |
|     |      | \(N\)  |     |     |     | T}$$ |         |         |
|     |      |        |     |     |     |      | \(N\)   |         |
|     |      |        |     |     |     | (    |         |         |
|     |      |        |     |     |     | m/s) |         |         |
+-----+------+--------+-----+-----+-----+------+---------+---------+
| 1   |      |        |     |     |     |      |         |         |
+-----+------+--------+-----+-----+-----+------+---------+---------+
| 2   |      |        |     |     |     |      |         |         |
+-----+------+--------+-----+-----+-----+------+---------+---------+
| 3   |      |        |     |     |     |      |         |         |
+-----+------+--------+-----+-----+-----+------+---------+---------+
| 4   |      |        |     |     |     |      |         |         |
+-----+------+--------+-----+-----+-----+------+---------+---------+
| 5   |      |        |     |     |     |      |         |         |
+-----+------+--------+-----+-----+-----+------+---------+---------+

# How does the weight of the hanging mass compare to the value for the centripetal force? 

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**Activity III.**

On the graph below plot mass of the hanging mass (M) versus r/T^2^ (on
the x-axis)

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# What pattern do you see when you made your graph? \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

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**Activity IV.** *The Mass of a planet*

The mass of the planets in our solar system is given in the table below.
The mass of Earth is 598 x 10^22^ kg, which is
5,980,000,000,000,000,000,000,000 kg (598 with 22 zeros after that).

  ------------------------------------ ----------------------------------
  Planet                               Mass (kg)

  Mercury                              330 x 10^22^

  Venus                                488 x 10^22^

  Earth                                598 x 10^22^

  Mars                                 642 x 10^21^

  Jupiter                              190 x 10^25^

  Saturn                               568 x 10^24^

  Uranus                               868 x 10^23^

  Neptune                              103 x 10^24^

  Pluto                                129 x 10^21^
  ------------------------------------ ----------------------------------

How do we know the mass of the planets? Is there a scale large enough to
hold a planet?

![](media/image4.jpeg){width="2.8194444444444446in" height="1.875in"}

The method astronomers use to determine the mass of the planet is
similar to the procedure used in Activity III.

In space whenever a body such as a moon or satellite is orbiting a
planet, the centripetal force is provided by the gravitational
attraction between the two bodies (*F~g~*) which is equal to

$$F_{g} = G\frac{M_{1}M_{2}}{r^{2}}$$

*M~1~* is the mass of the body in the center, *M~2~* the masses of the
satellite, *r* is the distance between them and *G* is a constant which
is equal to .498 m^3^/(d^2^ kg).

![](media/image5.jpeg){width="5.5in" height="4.4319444444444445in"}

If the gravitational attraction is the centripetal force
($G\frac{M_{1}M_{2}}{r^{2}} = \frac{M_{2}v^{2}}{r}$) then we can get the
mass of the center body by measuring the distance between the bodies and
the time for the satellite to make a complete orbit.

$$M_{1} = \frac{4\pi^{2}}{G}\frac{r^{3}}{T^{2}}$$

If we plot *M~1~* and $\frac{r^{3}}{T^{2}}$ then we should get a
straight line with a slope equal to $\frac{4\pi^{2}}{G}$.

![](media/image6.jpeg){width="5.075in" height="3.796527777777778in"}

Or you can determine the mass of a planet by multiplying
$\frac{r^{3}}{T^{2}}$ by $\frac{4\pi^{2}}{G} = \text{79}\text{.}3$.

Using the time it takes for the given satellites to revolve around the
planet and the distance between the satellite and the planet, determine
the mass of the following planets.

## Jupiter

+-------------+-------------+-------------+-------------+-------------+
| Satellite   | Distance    | Days for 1  | R^3^/T^2^   | Mass of     |
|             | from        | revolution  |             | Jupiter     |
|             | Jupiter (m) | (d)         |             |             |
|             |             |             |             | (kg)        |
+-------------+-------------+-------------+-------------+-------------+
| Io          | 422 x 10^6^ | 1.77        |             |             |
+-------------+-------------+-------------+-------------+-------------+
| Europa      | 671 x 10^6^ | 3.55        |             |             |
+-------------+-------------+-------------+-------------+-------------+
| Callisto    | 189 x 10^7^ | 16.69       |             |             |
+-------------+-------------+-------------+-------------+-------------+

## Mars

+-------------+-------------+-------------+-------------+-------------+
| Satellite   | Distance    | Days for 1  | R^3^/T^2^   | Mass of     |
|             | from Mars   | revolution  |             | Mars        |
|             | (m)         | (d)         |             |             |
|             |             |             |             | (kg)        |
+-------------+-------------+-------------+-------------+-------------+
| Phobos      | 940 x 10^4^ | 0.32        |             |             |
+-------------+-------------+-------------+-------------+-------------+
| Deimos      | 235 x 10^5^ | 1.26        |             |             |
+-------------+-------------+-------------+-------------+-------------+

## Earth

+-------------+-------------+-------------+-------------+-------------+
| Satellite   | Distance    | Days for 1  | R^3^/T^2^   | Mass of     |
|             | from Earth  | revolution  |             | Earth       |
|             | (m)         | (d)         |             |             |
|             |             |             |             | (kg)        |
+-------------+-------------+-------------+-------------+-------------+
| Moon        | 384 x 10^6^ | 27.3        |             |             |
+-------------+-------------+-------------+-------------+-------------+

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# How do the values you obtained for the mass compare to the values in the table?\_\_\_\_\_\_\_\_

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