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Because of the earth’s rotation, a person living on top of a mountain moves at a faster speed than someone at sea level. The mountain dweller’s clocks thus run ...

Question

Because of the earth’s rotation, a person living on top of a mountain moves at a faster speed than someone at sea level. The mountain dweller’s clocks thus run slowly compared to those at sea level. If the average life span of a hermit is 80 years, on average how much longer would a hermit dwelling on the top of a 3000-m-high mountain live compared to a sea-level hermit?

Because of the earth’s rotation, a person living on top of a mountain moves at a faster speed than someone at sea level. The mountain dweller’s clocks thus run slowly compared to those at sea level. If the average life span of a hermit is 80 years, on average how much longer would a hermit dwelling on the top of a 3000-m-high mountain live compared to a sea-level hermit?



Answers

How long would a day be if the Earth were rotating so fast that objects at the equator were apparently weightless?

So how fast would the earth have to be rotating so that masses on the equator would seem weightless? So we're going to say that the gravity is going to be equal to the centripetal acceleration. This would make us feel wait list. By the way, this is the relationship that would make us feel weightless. And then this wood evil, of course, b squared over the radius of the Earth s so we can say that the velocity is equal to two pi times the radius of the earth divided by the period of the earth. And so we can say that G equals V squared over r sum E on we can substitute for V and saythe v squared over r. The race of the earth would be equal to four pi squared times the radius of the earth divided by t squared. So that would be g and then the period. But simply be to pie times the square root of the radius of the earth, divided by the acceleration due to gravity on Earth's surface. So tea is going to be equal to to pie times the square root of 6.38 times 10 to the six meters divided by 9.8 meters per second squared. And this is giving us 5.7 times 10 to the third seconds or approximately 84 minutes. So essentially, what this means is that means that if our days were not 24 hours long, but on ly 84 minutes long, we would of course be rotating at a much faster rate. And if we were rotating at this rate, the everything on the that lives on the equator would actually feel weightless. And of course I would be extremely, it's not beneficial. But of course, this is a very this is not, ah, period that can be achieved by Earth. Of course, again, the days would be only 84 minutes, so this would be the period. That's the end of the solution. Thank you for watching

So the last day off 20 centuries is longer than the first day by 20 century. Times 0.1 2nd are essentially so, which is zero point your toe second. Now how do I get that? So in the problem, it say's that that day at the end of first century is one millisecond longer than the day at the start of the first century. So if we have ah, a century so the top one is the first day. Then let's say that zero. Then if the border matter is the last day, then the last day will be bigger by one millisecond, which is one times one divided by 1000 which gives us 0.0 zero one. Ah, second, So which tells us that yeah, uh, which tells us that at the end of 20th century, we just need to multiply that increase off seconds per century with the number of centuries that we have so that we can get rid of the centuries and we're left with the unit second, Now the average. If we want to calculate the average day, Ah, increase on the time span of 20 centuries, then we need to take the average off. Um, this time increased. Plus, we need to take the average of, uh, this increase because and in the beginning of 20th century, the time, let's say waas, whatever the time waas plus zero because there was no increase. Then it got increase by zero pines your two seconds, so the average is just half of it. So that's the average on Spann off 20 centuries. Um, so that that gives us 0.1 2nd So that means the average day during 20 centuries is the open 01 2nd longer than the first date since the on Since the increase Walker's uniformly that means tea will be accumulated. So total time t must be average increase in the length of day average increase. Ah, in the length off day. So in the length off day and we multiply that the number of days So that times, um, number off these. So how many days do I How do we have in a century? So every increase in the length off day is basically 0.1 seconds. We divide that by D because that's the, uh, thes in terms off D and then we convert. We convert the century from year toe day. So basically 3 65 0.25 ah, days is equal to and year, and we can multiply that with 2000 years with this 20 century. So as we see here that we need our final answer in seconds. That means we can get rid off days and years, which gives us 73 05 seconds, which is roughly equal toe two hours. So as we see here that, um, the total daily increasing time is a lot like two hours is really, um a big number. Thank you.

Well, let's suppose that the person is on the question of the Earth and he is standing on a bathroom scale to measure his weight when the person is back, please, because of the rotation of the Earth in the normal force exerted by the scale on the person is zero, which means that a normal force is equal to, uh, Zito. Well, let's apply. Newton's second long, then, UH, result in force is M G minus normal force. This force will produce, uh, an axle relation. And according to Newton's second Law, this result in force is equal to mass times on the axe relation, right. And here, normal force is equal to M G minus m times he well, since normal force is equal to zero deal for the M zero equals m g minus M times a right and M cancels from both sides here for acceleration equals she so acceleration. It's equal to x relation to you do, uh, gravity. Right now we no need, uh, these acceleration is equal to we swear. Divided by r e. The lari is the radius of the earth. And since we equals, do by times are e divided by uh, time PT aid so ex relation and becomes ex relation e becomes for my square for pi square times are e divided by ah di square and further G equals Seems ah, the strong is equal to Jean So g equals for my square times Radius of the earth divided by time petered off the earth square And from here time pt it off the earth equals di Impeded r equals fruit four by square times, Radius of earth divided by Jeanne Let's substitute values. Uh, this equals route for by square multiplied by six point three sevens. You don't multiply. Bite into the power six divided by 9.8 beer for a time. Period of birth equals 1.4 far.


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