As the earth rotates, what is the speed of (a) a physics student in Miami, Florida, at latitude 26 degrees

Video transcript

Hi, everyone in this practice problem, we are being asked to calculate the linear speed of the Sydney Opera House. We will have the latitude of the Sydney Opera House to be 33.85 degrees south and the mean radius of the earth to be 6371 kilometer. The options given for the linear speed of the Sydney Opera House are a 256 m per second. B 258 m per second. C 385 m per second and D 463 m per second. So we are given that the earth's radius R equals to 6371 kilometer. I'm gonna write that down first just like so, and I feel like it's gonna be easier for us to actually first draw the system of our problem. I'm going to draw the uh the earth right here where the earth is going to be rotating upon its axis. So I'm gonna draw the axis right here. And the Sydney Opera House is located at a latitude of 33.85° South. Which means that it will create an angle with the horizontal axis here Where the angle is going to be 33.85° just like. So, so the distance from the middle of the earth right here to the earth's surface or to the Sydney Opera House, which I will represent with a sphere This distance right? There is going to be the radius of the earth which is going to be 6370 71 km. What we are interested at is going to be looking at the trajectory at which the Opera house is undergoing a uniform circular motion. So in this case, because the earth is rotating upon its axis here, the Sydney Opera House will be undergoing a circular motion and the trajectory of that circular motion is going to be represented by uh this track right here just like so OK, so essentially the Sydney Opera House is going to be rotating upon the earth with this trajectory right here. And what we are interested at is to find what the distance from the earth's horizontal axis is to the Sydney Opera House, which I'm going to call that with B right here. So this distance right here because that is going to be essentially the radius at which the Sydney Opera House is undergoing the uniform circular motion. All right. So we are given the earth's radius. But what we want to find is the B or the actual radius that the Sydney Opera House is going through the uniform circular motion. So from the latitude or angle of the Sydney Opera House, which is 33.85 degrees, we can calculate the radius B of the circle using the formula of B equals R multiplied by cosine of 33.85 degrees, which is essentially the B is just a projection of the R. So this will then give us a value of B equals to 6371 kilometer multiplied by cosine of 0.85 degrees. So the B will then correspond to 5.291 times 10 to the power of minus three kilometer, which will equals to 5.291 times 10 to the power of minus six m. Now, in order for us to calculate the linear speed, which will be represented by this V right here, the linear speed will equals to B which is the radius of the actual vertical uniform motion uh multiplied by the angular speed which is going to be omega, we know what the B is, but we still are missing the omega. So in order for us to get the omega or the angular speed, we have to notice that the earth rotates on its axis relative to the sun once every 24 hours. That means that the T or the period is going to be hours. And the omega is going to equal to two pi divided by the period T and that will be two pi divided by hours. And I'm gonna convert this into radiant per seconds by multiplying it with one hour divided by 3600 seconds. And that will give us an omega value of 7.272 times 10 to the power of minus five radiance per seconds. Awesome. So now that we find the angular speed omega, we can actually substitute all of our values into our linear speed equation. So the B is going to correspond to 5.291 times 10 to the power of minus six m. The Omega is going to be 7.27, 2 times 10 to the power of -5 radiance per seconds. And they will actually give us a linear speed of the Sydney Opera House to be 385 m per second. So 385 m per second is going to be the answer to this practice problem which will correspond to option C. So option C with the linear speed of the Sydney Opera House being 385 m/s is going to be the answer to this particular practice problem. If you guys still have any sort of confusion, please make sure to check out our other lesson videos on similar topics and they'll be all for this one. Thank you?