 Let's try another example. How does the period of Mars compare to the period of Earth? The mass of the Sun is equal to 1.988 times 10 to the 30 kilograms. The distance from Mars to the Sun is equal to 2.111 times 10 to the 8 kilometers. We have previously derived an equation for the period of a satellite undergoing uniform circular motion, and this equation is T the period squared divided by r cubed is equal to 4 pi squared divided by Big G Big M. For this problem, we would like to find T, the period. So rearranging the equation above, we get that T is equal to 4 pi squared r cubed divided by Big G Big M, square rooted. But Big G is equal to the universal constant of gravitation. We know that r is equal to d is equal to 2.111 times 10 to the 8 kilometers, or 2.111 times 10 to the 11 meters. And we know that the mass of the Sun, which is the mass that is being orbited, is 1.988 times 10 to the 30 kilograms. Let's plug in these numbers. And this is equal to 5.29 times 10 to the 7 multiplied by the square root of meters over newtons, which are kilogram meters per second squared divided by kilograms. We cancel the kilograms, the meters, and the square of the second. We see that this is 1 over 1 over 1 second, which is equal to seconds, and therefore T is equal to 5.29 times 10 to the 7 seconds. What is this in days? If we multiply our answer in seconds by 1 minute over 60 seconds, which is the same as multiplying it by 1, and we multiply that by 1 hour over 60 minutes, and we multiply that by 1 day over 24 hours, we can convert it into days. Plugging this in, we get an answer of T equal to 612.53 days. So we have the period of Mars is equal to 612.53 days. We know that the period of Earth is equal to 365 days, so how does the period of Mars compare to that of Earth? If we divide the period of Mars by the period of Earth, we come out with a final answer of 1.68. Therefore Mars takes 1.68 times longer than Earth to orbit the Sun. Now if you look up the orbital period of Mars compared to that of Earth, you'll find that Mars actually takes 1.88 times longer to orbit the Sun. Have we done anything wrong? As with many exercises involving uniform circular motion, in this example we made the approximation that Mars was travelling in a circular orbit. In reality Mars travels around a slightly elliptical orbit as does Earth. However we're still very close to the correct answer, and you can see even with this approximation we're only about 10% off the correct answer. So in this problem we found the period of Mars and compared it to the known period of Earth, but is there any other way we could have approached this problem? Let's examine the equation we used to find the period of Mars. Rearranging for period gives us, you may notice when you look at this equation that it is independent of the mass of the object that is undergoing uniform circular motion. How can we use this? If we compare the period of Mars to the period of Earth, we see that most of the variables will cancel, and where left with the period of Mars is equal to the square root of the distance from Mars to the Sun cubed, divided by the distance from the Earth to the Sun cubed. We've already been given the radius of Mars's uniform circular motion, which is the distance from Mars to the Sun, which is equal to 2.111 times 10 to the 8 km. The distance from Earth to the Sun, which is the radius of Earth's orbit, is equal to 1.471 times 10 to the 8 km. If we plug in these values to compare the period of Mars to the period of Earth, we find that the answer is 1.72 to 2 decimal places. Using this method we found that the orbital period of Mars is equal to 1.72 times the period of Earth, which is very similar to our previous answer of 1.68. But why isn't the ratio exactly the same as we calculated before? Well, before, we just assumed that Mars's orbit was circular and used the experimentally measured value for the Earth's period. In the second calculation, we assumed that both the Earth and Mars have perfectly circular orbits. If we calculated the Earth's period assuming the orbit was circular, it'd be a bit less than 365 days.