Question

Two particles of masses m1 and m2 move uniformly in different circles of radii R1 andR2 about the origin in the x, y-plane. The coordinates of the two particles in meters are given as follows (z=0 for both). Here t is in seconds: x1(t)=4cos(2t)y1(t)=4sin(2t)x2(t)=2cos(3t−π2)y2(t)=2sin(3t−π2)

a. x1(t) = 4cos(2t)
b. y1(t) = 4sin(2t)
c. x2(t) = 2cos(3t- π/2)
d. y2 (t) = 2sin(3t- π/2)

Answer 1

The radii of the circles of motion for the two particles are R1 = 4 and R2 = 2. The x and y coordinates of the center of mass can be found using the formula XCM (t) = (m1x1(t) + m2x2(t))/(m1 + m2) and YCM (t) = (m1y1(t) + m2y2(t))/(m1 + m2). The center of mass moves in a circle.

Step-by-step explanation:

To find the radii of the circles of motion for the two particles, we need to use their x and y coordinates. From the given information, particle 1 has coordinates x1(t) = 4cos(2t) and y1(t) = 4sin(2t), while particle 2 has coordinates x2(t) = 2cos(3t - π/2) and y2(t) = 2sin(3t - π/2). The radii of the circles are R1 = 4 and R2 = 2, respectively.

To find the x and y coordinates of the center of mass, we can use the formula:

XCM (t) = (m1x1(t) + m2x2(t))/(m1 + m2)

YCM (t) = (m1y1(t) + m2y2(t))/(m1 + m2)

Using the given information, we can substitute the values and get the center of mass coordinates.

Finally, to determine if the center of mass moves in a circle or not, we can analyze the equations for XCM(t) and YCM(t) to see if they have any periodic behavior. In this case, both equations have periodic behavior, indicating that the center of mass moves in a circle.