Final answer:
The velocity and acceleration vectors of a particle moving in a circular path are the first and second derivatives of the position function, respectively. The acceleration vector pointing opposite to the position vector indicates centripetal acceleration. The centripetal force vector can be calculated by multiplying the mass by the acceleration vector.
Step-by-step explanation:
A particle with mass 0.50 kg moves in a circular path in the xy-plane, and its position is given with the vector equation r(t) = (4.0 cos 3t)i + (4.0 sin 3t)j. To find the velocity vector V(t), we differentiate r(t) with respect to t. This results in V(t) = -12.0 sin(3t)i + 12.0 cos(3t)j. To find the acceleration vector a(t), we differentiate V(t) with respect to t, giving us a(t) = -36.0 cos(3t)i - 36.0 sin(3t)j.
The acceleration vector points towards the center of the circle, which is a characteristic of centripetal acceleration. As the acceleration vector is directed opposite to the position vector, it confirms that the particle is experiencing centripetal force, which always points towards the center of the circular path.
To find the centripetal force vector as a function of time, we use the equation F(t) = m * a(t), where m is the mass of the particle. This results in F(t) = -18.0 cos(3t)i - 18.0 sin(3t)j N, considering that mass m is 0.50 kg.