Final answer:
To find the velocity and acceleration vectors as functions of time for a particle moving in a circular path, we differentiate the position vector with respect to time. The velocity and acceleration vectors are then determined by taking the derivatives. The acceleration vector always points towards the center of the circle, showing that it represents centripetal acceleration. The centripetal force vector is found by multiplying the particle's mass by the magnitude of the centripetal acceleration.
Step-by-step explanation:
To find the velocity and acceleration vectors as functions of time for a particle moving in a circular path, we can differentiate the position vector with respect to time. Let's call the particle's position vector r(t) = (4.0 cos 3t)i + (4.0 sin 3t)Ĵ. The velocity vector is found by taking the derivative of r(t) with respect to time. The acceleration vector is found by taking the derivative of the velocity vector with respect to time.
After taking the derivatives, we find that the velocity vector v(t) = -12.0 sin 3t i + 12.0 cos 3t Ĵ and the acceleration vector a(t) = -36.0 cos 3t i - 36.0 sin 3t Ĵ.
To show that the acceleration vector always points toward the center of the circle, we can calculate its dot product with the position vector. The dot product is given by a(t) · r(t) = 0, which means the acceleration vector and the position vector are orthogonal. Therefore, the acceleration vector always points toward the center of the circle.
The centripetal force is given by the mass of the particle times its centripetal acceleration. The centripetal acceleration can be found by taking the magnitude of the acceleration vector. So the centripetal force vector F(t) = m * |a(t)| = m * 36.0 √(cos^2(3t) + sin^2(3t)) = m * 36.0, where m is the mass of the particle.