Final answer:
To find the final position vector of an object moving in a circle after an 8.60 rad angular displacement, we calculate the final angle within one circle and then express the Cartesian coordinates using the radius and angle, resulting in (1.150 m)i + (2.443 m)j.
Step-by-step explanation:
To determine the final position vector of a small object moving in a circle after undergoing an angular displacement, we need to use polar coordinates. The object starts from a position vector of 2.70 m, and since it moves counterclockwise, the angular displacement will be added to the initial angle. The initial position vector at the starting point (when the angle is 0 rad) is 2.70 m.
The final angle (θ) after the displacement can be calculated as θ(initial) + angular displacement, which is 0 + 8.60 rad = 8.60 rad. However, this angle may be more than one full rotation (2π rad is a full circle). To find the actual angle within one full circle, we take the remainder when dividing by 2π rad.
θ(final) = 8.60 rad % (2π rad) = 8.60 rad % (6.28 rad) = 2.032 rad.
Since the object is on a circular path, its position vector in Cartesian coordinates can be represented as r = r(cos θ)i + r(sin θ)j, where r is the radius of the circle. Plugging in the radius (2.70 m) and the final angle (2.032 rad), we can calculate the components of the final position vector:
- x = r * cos(θ) = 2.70 m * cos(2.032 rad) = 2.70 m * 0.426 = 1.150 m
- y = r * sin(θ) = 2.70 m * sin(2.032 rad) = 2.70 m * 0.905 = 2.443 m
Therefore, the final position vector is (1.150 m)i + (2.443 m)j.