The existence of the planet Pluto was proposed based on irregularities in Neptune's orbit. Pluto was subsequently discovered near its predicted position. But it now appears that the discovery was fortuitous, since Pluto is small and the irregularities in Neptune's orbit were not well known. Answer the following to illustrate that Pluto has a minor effect on the orbit of Neptune compared with other planets.
(a) Calculate the acceleration of gravity at Neptune due to Pluto when they are 5.00 ✕ 1012 m apart. The mass of Pluto is 1.4 ✕ 1022 kg.
m/s2
(b) Calculate the acceleration of gravity at Neptune due to Uranus, when they are 2.50 ✕ 1012 m apart. The mass of Uranus is 8.62 ✕ 1025 kg.
m/s2
Give the ratio of this acceleration to that due to Pluto.
(acceleration from part (b) / acceleration due to Pluto)
yeet
(a) To calculate the acceleration of gravity at Neptune due to Pluto, we can use Newton's law of universal gravitation:
F = G * (m1 * m2) / r²
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.
First, let's calculate the force between Pluto and Neptune:
F = G * (m1 * m2) / r²
F = (6.67430 × 10^-11 N m²/kg²) * ((1.4 × 10^22 kg) * (1.02 × 10^26 kg)) / (5.00 × 10^12 m)²
Note: The mass of Neptune is approximately 1.02 × 10^26 kg.
Calculating this expression gives us the force between Pluto and Neptune.
Now, we can use Newton's second law, F = ma, to find the acceleration:
F = ma
a = F / m2
Here, m2 is the mass of Neptune.
Substituting the values, we get:
a = (F / m2)
a = ((6.67430 × 10^-11 N m²/kg²) * (1.4 × 10^22 kg) * (1.02 × 10^26 kg)) / (5.00 × 10^12 m)²
Calculate this expression to find the acceleration.
(b) To calculate the acceleration of gravity at Neptune due to Uranus, we can follow the same steps:
F = G * (m1 * m2) / r²
F = (6.67430 × 10^-11 N m²/kg²) * ((8.62 × 10^25 kg) * (1.02 × 10^26 kg)) / (2.50 × 10^12 m)²
Now, using Newton's second law:
a = (F / m2)
a = ((6.67430 × 10^-11 N m²/kg²) * (8.62 × 10^25 kg) * (1.02 × 10^26 kg)) / (2.50 × 10^12 m)²
Calculate this expression to find the acceleration.
Finally, to find the ratio of the acceleration due to Uranus to the acceleration due to Pluto, divide the value obtained in part (b) by the value obtained in part (a).
To calculate the accelerations of gravity at Neptune due to Pluto and Uranus, we can use the formula for gravitational acceleration:
a = G * (m1 / r^2)
Where:
a is the acceleration of gravity
G is the gravitational constant (6.67430 × 10^-11 N m^2 / kg^2)
m1 is the mass of the planet
r is the distance between the two planets
Let's start by calculating the acceleration of gravity at Neptune due to Pluto:
(a) Calculate the acceleration of gravity at Neptune due to Pluto when they are 5.00 ✕ 1012 m apart. The mass of Pluto is 1.4 ✕ 1022 kg.
Using the formula mentioned above, we have:
a_pluto = G * (m_pluto / r^2)
Substituting the given values:
G = 6.67430 × 10^-11 N m^2 / kg^2
m_pluto = 1.4 ✕ 10^22 kg
r = 5.00 ✕ 10^12 m
Plugging in these values, we get:
a_pluto = (6.67430 × 10^-11 N m^2 / kg^2) * (1.4 ✕ 10^22 kg) / (5.00 ✕ 10^12 m)^2
Now we can calculate it:
a_pluto ≈ 9.346 × 10^-6 m/s^2
So, the acceleration of gravity at Neptune due to Pluto is approximately 9.346 × 10^-6 m/s^2.
Moving on to the acceleration of gravity at Neptune due to Uranus:
(b) Calculate the acceleration of gravity at Neptune due to Uranus when they are 2.50 ✕ 1012 m apart. The mass of Uranus is 8.62 ✕ 10^25 kg.
Using the same formula, we have:
a_uranus = G * (m_uranus / r^2)
Substituting the given values:
G = 6.67430 × 10^-11 N m^2 / kg^2
m_uranus = 8.62 ✕ 10^25 kg
r = 2.50 ✕ 10^12 m
Plugging in these values, we get:
a_uranus = (6.67430 × 10^-11 N m^2 / kg^2) * (8.62 ✕ 10^25 kg) / (2.50 ✕ 10^12 m)^2
Now we can calculate it:
a_uranus ≈ 1.125 × 10^-5 m/s^2
So, the acceleration of gravity at Neptune due to Uranus is approximately 1.125 × 10^-5 m/s^2.
Now, to find the ratio of the acceleration due to Uranus to the acceleration due to Pluto, we divide the two values:
Ratio = acceleration_due_to_uranus / acceleration_due_to_pluto
Ratio = (1.125 × 10^-5 m/s^2) / (9.346 × 10^-6 m/s^2)
Ratio ≈ 1.20
Therefore, the ratio of the acceleration of gravity at Neptune due to Uranus to that due to Pluto is approximately 1.20. This implies that Uranus has a slightly greater effect on the orbit of Neptune compared to Pluto.