A particular asteroid has a circular orbit about the Sun with a radius of 450 million miles. How long does it take to go around the sun? Compare the prediction of Kepler's 2nd law to a direct computation with Newton's 2nd law. How much area does the line connecting the asteroid to the sun sweep out per time?
To determine the time it takes for an asteroid to go around the Sun and to compare Kepler's 2nd law with a direct computation using Newton's 2nd law, we'll need to consider the relevant formulas and principles.
1. Using Kepler's 2nd law:
Kepler's 2nd law states that the line connecting a planet or asteroid to the Sun sweeps out equal areas in equal time intervals. In other words, if we consider the imaginary line connecting the asteroid to the Sun, the area swept per time remains constant.
2. Using Newton's 2nd law:
Newton's 2nd law relates the forces acting on an object to its acceleration and mass. In this case, we can use the concept of centripetal force to calculate the time it takes for the asteroid to orbit the Sun.
Let's break down the problem step by step:
Step 1: Calculate the orbital period using Kepler's 2nd law:
Kepler's 2nd law states that the line connecting the asteroid to the Sun sweeps out equal areas in equal time intervals. Since the asteroid has a circular orbit, each complete orbit will correspond to an equal area swept. This area is a circle with radius equal to the orbital radius of the asteroid.
Using the equation for the area of a circle (A = π * r^2), we find that the area swept by the line connecting the asteroid to the Sun is A = π * (450 million miles)^2.
To find the time it takes for the asteroid to complete one orbit, we need to divide the total area swept by the area swept per unit time. However, we need additional information to compute the area swept per unit time.
Step 2: Calculate the orbital period using Newton's 2nd law:
To determine the orbital period using Newton's 2nd law, we need to relate the gravitational force between the Sun and the asteroid to the centripetal force required to keep the asteroid in its orbit.
The gravitational force is given by Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
The centripetal force is given by:
F = (m * v^2) / r
Since the gravitational force and the centripetal force are equal, we have:
G * (m_Sun * m_asteroid) / r^2 = (m_asteroid * v^2) / r
G * m_Sun / r = v^2
v = sqrt(G * m_Sun / r)
The orbital period (T) is given by T = (2 * π * r) / v.
Step 3: Compare the predictions of Kepler's and Newton's laws:
To compare the predictions, calculate the time it takes for the asteroid to orbit the Sun using both Kepler's 2nd law and Newton's 2nd law. If the predictions match, it confirms their agreement.
Step 4: Calculate the area swept per unit time:
To determine the area swept per time by the line connecting the asteroid to the Sun, we need to calculate the rate of sweeping. This can be obtained by dividing the total area swept by the orbital period from either Kepler's or Newton's law.
By following these steps, you can find the time it takes for the asteroid to orbit the Sun, compare Kepler's and Newton's laws, and determine the area swept per unit time.