Download Question (1340001) Solution

Question (1340001)
The gravitational force by a star on a planet.
A planet of mass 4.0 × 1024 kg is at a location < 5.0 × 1011 , −2.0 × 1011 , 0 > m. A star of mass 5.0 × 1030 kg
is at a location < −2.0 × 1011 , 3.0 × 1011 , 0 > m. What is the gravitational force by the star on the planet?
Solution
Begin by drawing a picture, showing the star and planet, and their positions.
Figure 1: A star and planet.
To calculate the gravitational force on the planet, we need the relative position of the planet (i.e. the position
of the planet relative to the star). This is the position vector from the star to the planet.
Figure 2: The relative position vector of the planet.
The gravitational force on the planet is given by Newton's Law of Gravitation.
F~grav
by star on planet
= −G
mplanet mstar
r̂
|~r|2
where ~r is the relative position of the planet. The relative position of the planet in this case is
~r
= ~rplanet − ~rstar
= < 5 × 1011 , −2 × 1011 , 0 > m− < −2 × 1011 , 3 × 1011 , 0 > m
= < 7 × 1011 , −5 × 1011 , 0 > m
Note that the relative position vector points to the right and downward which matches the picture, giving
us condence that our calculation is correct. Find the magnitude of ~r and its unit vector.
|~r| =
=
r̂
q
2
(7 × 1011 m) + (−5 × 1011 m)
2
8.6 × 1011 m
~r
|~r|
< 7 × 1011 , −5 × 1011 , 0 > m
=
8.6 × 1011 m
= < 0.814, −0.581, 0 >
=
Now, substitute the given quantities and solve for the gravitational force on the planet.
F~grav
by star on planet
= −G
mplanet mstar
r̂
|~r|2
= −(6.7 × 10−11 N m2 /kg2 )
=
(4 × 1024 kg)(5 × 1030 kg)
(8.6 × 1011 m)
2
< 0.814, −0.581, 0 >
(1.811 × 1021 N) < −0.814, 0.581, 0 >
= < −1.5 × 1021 , 1.1 × 1021 , 0 > N
Note that the gravitational force points to the left and upward. This is the direction from the planet toward
the star, which is consistent with the fact that gravitational force is attractive. A picture of the gravitational
force on the planet is shown below. Note that the direction of the gravitational force on the planet is to the
left and upward, which is consistent with our calculation.
Figure 3: The gravitational force by the star on the planet.