Calc. speed & time in a uniform circular motion - Solved numerical

- Vinayak drives in a uniform circular motion of radius seven kilometers. If it takes two hours to complete one round, find his speed. It's given to take pi as 22 over seven. So let's think about what's given to us. We're given that Vinayak is driving in, I don't know which way, but he's driving in a uniform circular motion of radius seven kilometers. So let's say he's driving a car on his circular track. Then the radius of this track, the radius of this circular track, is given to be seven kilometers. Seven kilometers. And it takes him two hours to complete one round. So, for him to complete one entire round, it takes him two hours. So the time for that, so let me just write that as two hours, it takes him two hours for one round. What we need to calculate is the speed. Now the important thing to notice over here is given that Vinayak is driving in a uniform circular motion. What does that mean? What is the meaning of uniform circular motion? It just means that the speed of this car is a constant. As it goes over the entire circular part, its speed does not change, its speed is a constant. And because the speed is a constant, we can just go ahead and use speed equals distance over time. That's how we always calculate speed, right? Provided speed is a constant. If speed is changing, then we can't do this, because then this number keeps changing, isn't it. So, let's see what the distance is, and let's see what the time is. We have to calculate what the speed is, right? So what is the distance traveled? In one round, in one complete round, what is the distance traveled. Well, the distance is this path, this path length, and that is the circumference of the circle, right? So the distance traveled is this circumference. And how do we calculate the circumference of any circle? We calculate it as two pi r. That's where the pi comes. And for that one entire round, we know the time. The time taken is two hours. So let's just write that down. Time is two hours. And of course since the units are not in standard, we have hours and we have kilometers, we need to be careful, we'll just put the numbers with the units. And so if we substitute, let's write that down over here, speed equals two pi. They ask us to take pi 22 over seven. It's not exactly 22 over seven, but they've asked us to do that so the calculations become simpler. Times the radius. The radius is seven kilometers, right, with the units. Divide by two hours. And let's see how much we get. We get, this cancels, and this cancels. So we end up with 22, right, that's all. Twenty-two, that's it. So that's the speed. The speed of our vehicle is going to be 22 kilometers per hour. All right, let's do one more. An ant is traveling in a uniform circular motion with a speed of 11 meters per second. If the radius of that circle is 21 meters, find the time taken to cover half a round. So a very similar problem, but a slight difference, right? The difference over here is we're asked to calculate the time taken. We're given the speed and time taken to cover half a round. So, can you try this? Go ahead, give it a shot. Okay, again, let's make a drawing. I'm going to do the same drawing as before. So let's imagine now this is our ant. It's given that the radius of the circle is 21 meters. So this time this radius becomes 21 meters. And we are given the speed, we know the speed of this ant. So the speed of that ant is 11 meters per second. And we are asked to calculate the time taken. How much time it takes to cover half a round. That means we should only think about half a round, so that's from here all the way to here, that's it. Only from here to here. Okay, so again, because it's a uniform circular motion, the speed is a constant. So we can just go ahead and use speed equals distance over time. Distance over time. And since time is asked, well let's rearrange the equation. If we rearrange we get time equals distance over speed. Distance over speed. And speed is given to us. We can figure out what the distance is. We should only think about the distance for half a round, because only for half a round I need to calculate the time. So again, if you couldn't do this earlier, no problem. Now again, can you see if you can give it a shot. Good, give it a shot. All right, so if we substitute, what is the distance? Since we are only now concerned for half a round, the distance will be half the circumference. A full circumference is two pi r, so half the circumference will be just pi r. Right, divide by 2, isn't it? Two pi r is full circumference, so half of that is just pi r. Makes sense, right? Divide by speed, is given as 11 meters per second. Now everything is in SI, we don't have to use the units, but it's okay, let's put in the units no problem. So time will be pi, what shall we do for pi? Now I should have mentioned in the problem, pi is 22 by seven. So here also, let's take it to be 22 by seven. Assume that it's given in the problem itself, even though I have not written that. So, pi is 22 by seven. Times r, r is 21 meters. Divided by 11 meters per second. Okay, we have some fractions in the numerator, so let's be very careful over here. Ah, let's see if something cancels out. Yes, seven goes one time, seven goes three times. So, this will be, we get 22 times three, I will not multiply, because I have to do another division. Divide by 11 meters per second. Eleven goes two times, and I end up with six. Six. And the meter cancels, and the one over second becomes second. So I end up with six seconds. And so, the ant takes six seconds to complete half a round. So what we see is whenever we are dealing with uniform circular motion, we can always use speed is distance by time, going back to the old formula. And, the only thing is, when it comes to the distance, we just have to use the circumference formula. If it's a full round it will be two pi r, if it's a half a round, it will just be pi r.