Two particles of masses m1 and m2 move uniformly in different circles of radii R1 and R2 about o

step 1 So once again we have two particles and these particles. This one is M1 and then we have a secon

Two particles of masses m1 and m2 move uniformly in...

Key Concepts

Uniform Circular Motion

Uniform circular motion describes the motion of an object traveling at a constant speed along a circular path. Its position is typically expressed in parametric form using trigonometric functions, where the x and y coordinates are given by x = R cos(?t) and y = R sin(?t), with R representing the radius of the circle and ? the constant angular speed.

Center-of-Mass Calculation

The center of mass of a system of particles is the weighted average of the positions of all the particles, where each particle's position is weighted by its mass. For a two-particle system, the center-of-mass coordinates are determined by the equations x_CM = (m1*x1 + m2*x2) / (m1 + m2) and y_CM = (m1*y1 + m2*y2) / (m1 + m2), allowing one to relate the individual particle positions to the overall system behavior.

Relative Motion of Particles

When analyzing the motion of particles relative to their center of mass, the positions of individual particles can be expressed as deviations from the center-of-mass coordinates. This concept is crucial when the movement data of one particle and the center of mass are known, as it enables the determination of the unknown particle's position through vector relations and the balancing of momentum contributions from both particles.

Transcript

00:01 So once again we have two particles and these particles.

00:11 This one is m1 and then we have a second one which is m2.

00:16 Obviously m1 is going to have its own radius r1 and then m2 is also going to have its own radius r2.

00:29 And assume that these two particles are sitting on the origin.

00:39 So this is relative, so assume that actually we need to backtrack that a little bit.

00:49 So we have two particles and these two particles, you know, we have m1 and m2 and relative to the origin.

01:00 They have different radiuses so these radiuses are one and this radiuses are two and the motion the motion is uniform meaning that the acceleration is constant acceleration that's uniform motion and it so happens that the two objects you know have a central of mass.

01:35 If you think about the coordinates, the x, y, this particle one right here, if we're looking at this particle, it's based off of specific coordinates and the x coordinate for particle is the same as 4 cosine of 2t and the y coordinate of the same particle is the same as for a sign of 2t.

02:11 Remember, this is the y direction and this is the x direction.

02:16 And the second thing that happens is these two objects have a center of mass.

02:22 You know, we're assuming the two objects have a center of mass, and that center of mass for these two objects in the x direction is we also have, is given us 3 cosine of 2t and also in the wide direction, the center of mass is given us 3 sign of 2t.

02:48 So those are the pieces of information that were given in the problem.

02:53 And all of these, the lengths are in meters and the time is in seconds, obviously, because this is a uniform motion.

03:06 So the first part, we want to find r1.

03:12 So this is the radius of motion for particle 1.

03:21 And then we also want to find the x and y coordinates of particle 2.

03:31 So in other words, we're saying x2t and y2t, their coordinates.

03:39 And we also want to find the radius of that.

03:45 If you think about the radius of a circle, this is the radius of a circle.

03:52 We want to make sure it's centered.

03:55 Circle, so it's centered right there.

03:58 Assuming this is the radius, you're going to see that there is a triangle with the x and the y direction.

04:06 And so the radius becomes radical x squared plus y.

04:12 Squared and for the center of mass the expression is m1 velocity 1 plus m2 velocity 2 all of m1 plus m2 just to recall we want to find the radius of 1 also the radius of 2 and the coordinates of particle 2 that's what we're looking for so the radius of particle 1 is it's going to be the same as the x coordinate, which is 4 cosine of 2t squared, plus the y coordinate, which is 4 sine of 2t squared, you want to square that, the x and the y coordinate.

05:04 Assuming that x1 of t was the same as 4 cosine of 2t, and y1 of t was the same as 4 sine of 2t.

05:17 So we simplify this and it becomes four squared, cosine squared of 2t plus 4 squared sine squared of 2t.

05:32 Obviously this is going to simplify too.

05:35 We can pull out the 4 squared and inside we have cosine squared 2 t, sine squared of 2t.

05:45 Recall that sine squared theta plus cosine squared data is the same as 1.

05:54 So this is same as 4 squared times 1.

05:59 And 4 squared is 16.

06:02 So this is radical 16.

06:04 And when it comes out, it's plus minus 4.

06:07 But in terms of radius, we can't really have the negative one.

06:12 So the radius are 1 becomes the same.

06:16 As positive 4 meters.

06:20 That's the same as r1.

06:22 It's positive 4 meters.

06:24 And the second part of the problem was supposed to figure out the coordinates of r2 and also the radius of r2.

06:39 The center of mass is given us xm is m1, x1 plus m2, x2, all of m1 plus m2 so we want to reorganize this problem such that we solve for x2 that's going to be the same as m1 plus m2 xcm minus m1 x1 all of m2 so all we did is we multiply both sides by m1 plus m2 and then we subtracted this one so we subtracted m1 x1 we subtracted m subtracted m1 x1 on both sides then of course we're left with m2 and divided both sides by m2 so that's what happened but both sides by m2 ended up with a new equation for x2 assuming that x1 t is 4 cosine of 2 t and x1 y1 t is the same as 4 sine of 2t.

08:03 We don't really need y1 at the moment because we're looking for x2 and we're going to say m1 plus m2.

08:11 The center of mass is 3 cosine of 2t.

08:17 You can recall if you go back, you see that the center for x is 3 cosine of 2t.

08:26 That's the center, so we're using that in the equation and then subtract m1.

08:32 X1 is 4 cosine of 2t, then we divide both sides by, or we divide the whole thing by m2...

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