Final answer:
The particle's circular motion is characterized by a constant speed Aω and a centripetal acceleration (âc) pointing towards the center. Velocity and acceleration are determined by differentiating the position vector. The centripetal force is mass times centripetal acceleration, Fc = maâc.
Step-by-step explanation:
The movement of a particle in circular motion can be thoroughly understood by analyzing its position, velocity, and acceleration vectors. Initially, the particle's position vector r(t) is given by r(t) = Acos(ωt)î + A sin(ωt)ð, where A is the radius of the circle, ω is the constant angular frequency, and t is the time. This equation indicates that the particle's path is circular with radius A.
To find the velocity, differentiate the position vector with respect to time. According to the given hints, the derivatives of the trigonometric functions are (d/dt)(cosωt) = -ωsinωt and (d/dt)(sinωt) = ωcosωt. The velocity vector can be calculated as dr/dt which, after applying these derivative rules, gives us the components of velocity in the î and ð directions.
The speed of the particle is constant and given by Aω, which is shown by computing the magnitude of the velocity vector obtained after differentiation.
To determine the acceleration, we differentiate the velocity vector again with respect to time using the derivative rules, which gives us the acceleration vector with components along the î and ð directions. Once calculated, it becomes apparent that the acceleration (â) is centripetal, pointing towards the center of the circle, and is given by âc = rω², where r is the radius.
Finally, we calculate the centripetal force, which is the net force required to keep the particle moving in a circle. According to Newton's second law, this force is the product of the mass (m) of the particle and its centripetal acceleration, thus Fc = mâc.