Earth's orbit around the Sun is nearly circular. The period is 1 yr =365.25 d. (a) In an elapsed

Earth's orbit around the Sun is nearly circular. The period...

Key Concepts

Average versus Instantaneous Acceleration

The average acceleration over a finite time interval is the total change in velocity divided by the time interval, while the instantaneous acceleration is the acceleration at a specific point in time, such as the centripetal acceleration in circular motion. Comparing these helps illustrate how the overall change in velocity over a day relates to the constant inward pull experienced at any moment in Earth's orbit.

Centripetal (Radial) Acceleration

Centripetal acceleration refers to the instantaneous acceleration that points toward the center of the circular path. It is responsible for changing the direction of the velocity vector in circular motion and is mathematically expressed as a = v²/r or, equivalently, as a = r?². This concept is essential for understanding the forces acting on Earth in its nearly circular orbit around the Sun.

Angular Displacement and Angular Velocity

Angular displacement is the angle through which an object moves on a circular path over a given time interval, and angular velocity is its rate of change. In problems involving orbits, the total angle of 2? radians for a complete orbit (over one year in Earth's case) allows one to compute the angular displacement for shorter time periods, such as one day, by proportional division.

Circular Motion

Circular motion describes the movement of an object along a circular path. It involves constant speed but continuously changing velocity due to the change in the direction of motion. This concept is fundamental because the Earth's orbit around the Sun is nearly circular, which means that even though the speed is approximately constant, the changing direction produces an acceleration towards the center of the circle.

Vector Change in Velocity

Even if the speed of an object in circular motion remains constant, its velocity – a vector quantity with both magnitude and direction – changes continuously. The change in velocity during a time interval is determined by differences in direction, and this vector change is particularly important for calculating average accelerations over finite time intervals in orbital motion.

Transcript

00:02 In this example, we're going to be looking at the earth as it orbits the sun, and we're going to try to determine some properties of the rotation of the system.

00:13 In particular, we're going to look at the angular displacement after one day.

00:20 We're going to look at the magnitude of the change in velocity of the earth, its average acceleration, and its instantaneous acceleration, and we're going to compare those values.

00:29 So to start off, we want to look at what the angular displacement is, delta theta in one day.

00:43 So to do this, we just have to remember that delta theta is related to the angular velocity of an orbit times the time interval.

01:02 And we can also recall that omega is related to the period of an orbit by 2 pi over the period.

01:19 So we can do a little trick here and notice that we have a delta t over a t.

01:27 And if we both, if we use units of days for this calculation for both delta t and t, we won't actually have to convert them because the units will just cancel out.

01:36 I'll show you what i mean down here.

01:38 So for the calculation, delta theta equals 2 pi times delta t over t, delta t is just one day.

01:51 And the period is 365 .25 days.

02:00 So since these were both in days, we actually don't have to do any conversions, which is nice.

02:07 So that comes out to about 0 .172 radiance or 1 .72 times 10 to the minus 2 radiance, which is slightly less than a degree.

02:40 So it's a very small angular displacement.

02:44 So now we're going to check out what the magnitude of the change in velocity is.

02:51 So remember as we go from one point of our orbit here to another, the orbit looks something like that.

03:04 The velocity here at the first point will be tangential to the curve.

03:11 If i'm lucky, i can draw a tangent line here, something like that.

03:15 And then this one will be a slight, we'll have the same magnitude, but it will be slightly rotated.

03:24 So if i were to move this arrow here up to the next one to try to look at the resultant vector, might look something like that.

03:37 And we expect the net resultant of these two vectors to be pointing in this direction.

03:46 So we're looking for the magnitude of that.

03:52 So the easiest way to calculate this one is to record.

03:55 That the magnitude of delta v is equal to v times delta theta, which is kind of an intermediate calculation we found while learning about circular motion.

04:12 So v here is going to just be 2 pi r over omega t, which we can find by remembering our or v's relationship to omega when we're dealing with uniform circular motion.

04:29 So v equals r omega, and then then again we remember that omega can be expressed in terms of the period, 2 pi over t.

04:40 So we just substitute that in for v, since we know r and we know t.

04:46 Or we can readily find r, since we can go look up what the distance from the sun to the earth is.

04:56 So the magnitude of delta v should be equal to b2 pi r over t times delta theta.

05:12 Now for r, just do a quick internet search or it's a constant that is likely in your textbook as well, but the distance to the, or from the sun to the earth, is about 1 .5 times 10 to the 11th meters.

05:30 And for our period now, we will actually have to convert this into seconds.

05:37 So our period is 365 .2 .25 days.

05:44 And we can remember that there are 24 hours in a day and there are 3 ,600 seconds in one hour.

06:06 So all those units should cancel out.

06:10 We're just left with the period there.

06:13 So if we substitute these values into our equation here and do a little number crunching, we will end up with magnitude of b.

06:28 It's about 514 meters per second.

06:36 Awesome.

06:39 So now let's try to find the average acceleration during this time.

06:53 So when the acceleration is not constant, we say that delta v over delta t, right, the change of velocity over the change in time, is the average acceleration during that time period.

07:07 If the acceleration is constant, then this is actually exactly the acceleration...

Please give Ace some feedback