Final answer:
The radius of the circle of motion for both particles is 8 units. The x-coordinate of the center of mass is 0 and the y-coordinate of the center of mass is 4. The center of mass does not move in a circular path.
Step-by-step explanation:
The parametric equations given are:
x = t²
y = t + 4
To find the radii of the circles of motion for both particles, we need to consider the range of values of t. The given range is -4 ≤ t ≤ 4. By substituting the values of t into the equations, we can find the corresponding x and y coordinates. The radii of the circles of motion will be the maximum distance from the center to any point on the curve for each particle.
For the first particle:
When t = -4, x = (-4)² = 16 and y = -4 + 4 = 0
When t = 4, x = (4)² = 16 and y = 4 + 4 = 8
The distance between these two points is √((16-16)² + (8-0)²) = 8. Therefore, the radius of the circle of motion for the first particle is 8 units.
For the second particle:
When t = -4, x = (-4)² = 16 and y = -4 + 4 = 0
When t = 4, x = (4)² = 16 and y = 4 + 4 = 8
The distance between these two points is √((16-16)² + (8-0)²) = 8. Therefore, the radius of the circle of motion for the second particle is 8 units.
To find the x- and y-coordinates of the center of mass, we need to find the average values of x and y for all values of t within the given range. By integrating x and y with respect to t and dividing by the length of the range, we can find the center of mass coordinates:
The x-coordinate of the center of mass = (1/(4-(-4))) * ∫[(t²) dt] from -4 to 4
The y-coordinate of the center of mass = (1/(4-(-4))) * ∫[(t+4) dt] from -4 to 4
By integrating the respective expressions and performing the calculations, we find that the x-coordinate of the center of mass is 0 and the y-coordinate of the center of mass is 4. Therefore, the center of mass is located at (0, 4).
To decide if the center of mass moves in a circle, we need to examine how the x- and y-coordinates of the center of mass change with respect to t. Since the x-coordinate of the center of mass is a constant 0, it does not change with t. However, the y-coordinate of the center of mass is always 4, which means the center of mass remains at a fixed y-coordinate, indicating that it does not move in a circular path.