Communications satellites are placed in a circular orbit where they stay directly over a fixed point on the equator as the earth rotates. These are called geosynchronous orbits. The radius of the earth is 6.37 x 10⁶ m, and the altitude of a geosynchronous orbit is 3.58 x 10⁷ m (≈ 22,000 miles). What are (a) the speed and (b) the magnitude of the acceleration of a satellite in a geosynchronous orbit?

Video transcript

Welcome back everybody. We are taking a look at a weather satellite that is taking part in a geo stationary orbit. Now, what does this mean here? Well, I'm going to represent this little circle as earth and I'm gonna put our satellite right up here now, since it has a geo stationary orbit, this means that it has a similar orbit or stays amongst the same position over earth as it orbits. Meaning its period of orbit is going to be the same as earth's which is 24 hours. We're told that it is resting at 36, kilometers above earth's surface. And we are also told that the radius of earth is so 6378 kilometers. And we are tasked with finding two things here. We are tasked with finding what the tangential velocity is of our satellite. And what is is acceleration as well. Luckily, we have formulas for both of these things are velocity is given by two pi R over the period and the acceleration is given by our velocity squared over our radius of rotation. Wonderful. So and I just let me go back here. This is also gonna be our radius of rotation. Let's find our velocity first and then we will be able to find our acceleration as well. Our velocity is equal to two pi times our radius of rotation. Now, here's the thing, we are rotating about the center of the earth. So our radius of rotation is going to be the sum of these two values right here. When you add them together, you get 42,378 kilometers. But we do want meters. So I'm going to multiply this by 10 to the third. Since there are 1000 m in a kilometer, this of course will be divided by our period of hours. But we want our period to also be represented in seconds in one hour. We know there's 3600 seconds. This will cancel out the unit of our. And if you multiply straight across, you will get our period in seconds, plugging all of this into our calculator. We then get a period of three point oh eight times 10 to the third meters per second. Great. So now with that, let's go ahead and find our acceleration. We have our velocity of 3.8 times 10 to the third squared divided by R radius of orbit which is once again 42,378 times 10 to the third. This then gives us an acceleration of 0.22 m per second. Squared. So now we have found both our velocity and acceleration of our satellite above earth and this corresponds to our final answer. Choice of D Thank you all so much for watching. Hope this video helped we will see you all in the next one.