Question

Two particles of masses m1 and m2 move uniformly in different circles of radii R1 and R2 about origin in the x, y-plane. The x- and y-coordinates of the center of mass and that of particle 1 are given as follows (where length is in meters and t in seconds): x1(t)=4cos(2t),y1(t)=4sin(2t) and: xCM(t)=3cos(2t),yCM(t)=3sin(2t).

a) 4:9
b) 9:4
c) 1:1
d) 2:3

Answer 1

a. The radius of the circle in which particle 1 moves is 4 meters. b. The x- and y-coordinates of particle 2 are 2cos(2t) and 2sin(2t) respectively. The radius of the circle for particle 2 is also 4 meters.

Step-by-step explanation:

a. To find the radius of the circle in which particle 1 moves, we can use the given x1(t) and y1(t) equations. The radius will be the distance from the origin to any point on the circle, which can be found using the Pythagorean theorem: R1 = sqrt(x1(t)^2 + y1(t)^2). Substituting the given equations, R1 = sqrt((4cos(2t))^2 + (4sin(2t))^2) = sqrt(16cos^2(2t) + 16sin^2(2t)) = sqrt(16) = 4 meters.

b. To find the x- and y-coordinates of particle 2, we can use the given xCM(t) and yCM(t) equations, since the center of mass coordinates are the average of the two particles' coordinates. Therefore, x2(t) = 2xCM(t) - x1(t) = 2(3cos(2t)) - 4cos(2t) = 2cos(2t) and y2(t) = 2yCM(t) - y1(t) = 2(3sin(2t)) - 4sin(2t) = 2sin(2t). The radius of the circle for particle 2 will be the same as the radius of the circle for particle 1, which is 4 meters.