(a) Calculate the magnitude of the angular momentum of the...

Key Concepts

Angular Momentum

Angular momentum is a measure of the quantity of rotation of an object and is analogous to linear momentum in translational motion. In physics, it is defined for particles using the cross product of the position vector and linear momentum (L = r × p), and for rigid bodies in rotational motion it is given by the product of the moment of inertia and the angular velocity (L = I?). This concept is crucial when analyzing both orbital motions and rotations of bodies about an axis.

Angular Momentum in Circular Orbits

When a body is in a circular orbit, it is often modeled as a particle to simplify the problem, assuming that its dimensions are negligible compared to the orbital radius. In this context, the angular momentum can be computed using the formula L = mvr or, equivalently, L = mr²?, where m is the mass of the object, v is the orbital speed, r is the radius of the orbit, and ? is the angular velocity. This approach is typically reasonable when the spatial extent of the orbiting body does not significantly affect its motion.

Moment of Inertia of a Uniform Sphere for Rotational Motion

When calculating the angular momentum due to rotation about an axis through the center of mass, as in the case of Earth rotating about its poles, it is modeled as a uniform solid sphere. The moment of inertia for such a sphere is given by I = (2/5)MR², where M is the mass of the sphere and R its radius. This parameter is key in determining the angular momentum (L = I?) associated with the rotational motion of extended bodies.

Transcript

00:01 In this question, we are asked to calculate the magnitude of the angular momentum of the earth in a circular orbit around the sun.

00:10 And we are asked to calculate the magnitude of the angular momentum of the earth due to its rotation around its axis through the north and south poles.

00:20 So this is two different types of angular momentum.

00:24 In the first part, part a, we're going to be thinking of the sun as a point.

00:31 Point particle.

00:32 And the reason why it is reasonable to do so is because the earth is very, very small in comparison to its orbit around the sun.

00:45 So if we're looking at the sun earth system, the earth basically appears as a point particle, and we can sort of ignore the fact that it has actual dimension.

00:59 So to calculate the angular momentum for a point particle, we're going to use mv cross r.

01:08 And essentially, what this boils down to for a circular orbit is just mvr.

01:16 So v cross r would technically be vr sign data, but v and r are at 90 degrees to one another for a circular orbit.

01:27 It.

01:30 And so the angle between them is 90.

01:34 And so sign of 90 is going to equal one and we don't need to include it.

01:39 So we're just going to use angular momentum is equal to mvr.

01:45 And we don't really know what the speed of the earth is, but what we can do is rewrite v as 2 pi r over the period.

02:01 So that's the distance that the earth travels around the sun, that's the circumference of its path to pi r, divided by the period or the time it takes to do a full circular orbit around the sun.

02:19 So overall, we get the following formula.

02:26 So we're going to just plug in our variables.

02:29 So the m will be the mass of the earth.

02:37 The r is going to be the radius of the orbit or the d.

02:41 Distance between the sun and the earth.

02:44 So that's going to be 149 .6 times 10 to the 9 meters squared.

02:56 And then that's going to be over the period.

02:59 We know that the period is 365 days.

03:02 It takes a year for the earth to complete one orbit around the sun.

03:08 But we're going to convert that into seconds because that's our standard unit.

03:14 So 365 days times 24 hours in a day times 60 minutes and an hour times 60 seconds in a minute.

03:27 That will get us our period in seconds.

03:31 So once we accurately enter all that into our calculator, that's going to give us 2 .66 times 10 to the 40, kilograms meters squared per second...

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