Radial and transverse acceleration

Question

A point p describes a curve r= ae^(theta) with a constant angular velocityabout its pole Find the radial and transverse acceleration of point P

Answer 1

The radial acceleration is the second derivative of r wrt t. You will use the chain rule for this one.

The tangential acceleration is the second derivative of θ wrt t. However we are told that the point/object moves with constant angular velocity. So we can write θ = ω t + c, and dθ/dt = ω = constant, the derivative of a constant is zero, so the tangential acceleration is zero.

dr/dt = (dr/dθ)(dθ/dt) (chain rule)

dr/dθ = d(a e^θ)/dθ = a e^θ and dθ/dt = ω from before

so dr/dt = a ω e^θ

and d²r/dt² = aω (d(e^θ)/dθ)(dθ/dt) = a ω² e^θ but a e^θ = r so d²r/dt² = ω² r, which is the radial acceleration (centripetal acceleration)