Final answer:
The curvature for the circle described by r(t) can be found using the curvature formula κ = |r'(t) × r''(t)| / |r'(t)|^3, and angular velocity is calculated with ω = ∆ϴ/∆t. Total acceleration combines centripetal and tangential acceleration, while the radius of curvature is dependent on the circular path's geometry or derived from physical measures in dynamics.
Step-by-step explanation:
To find the curvature for the circle given by the vector function r(t) = 2cos t i + 2sin t j, we use the formula for curvature (κ) of a curve at a particular point, which is defined as κ = |r'(t) × r''(t)| / |r'(t)|^3. In this context, r'(t) and r''(t) represent the first and second derivatives of the vector function r(t) with respect to time (t), respectively.
The formula for angular velocity, ω, is given by ω = ∆ϴ/∆t, where ∆ϴ is the change in rotation angle and ∆t is the change in time. The radius of curvature (r) is simply the radius of the circular path the object is moving on.
Given that the acceleration vector a(t) has two components, the centripetal acceleration towards the circle's center and the tangential acceleration which is the derivative of the speed with respect to time, we can find the total acceleration by combining these two components.
The radius of curvature can often be found using the formula r = mv/qB, where m is the mass, v is the velocity, q is the charge, and B is the magnetic field strength. However, in kinematics, the radius of curvature is directly obtained from the circular path's geometry.