2) The position of a particle is given r(t) = A(cos wt i + sin wt f), where w is a constant. (a) Show that the particle moves in a circle of radius A. (b) Calculate and then show that the speed of the particle is a constant. (c) Determine and show that a is given by ac = rw. (d) Calculate the centripetal force on the particle.

Explanation:

r(t) = A(cos wt i + sin wt f)

= A cos wt i + A sin wt j

x = A cos wt

y = A sin wt

radius r

r² = x² + y² ( This is equation of a circle with radius r  )

=  A² cos² wt +A² sin² wt

= A²

r = A

radius r = A

b )

speed = dr/dt

v = - Aw sinwt i + Aw coswt j

magnitude of velocity

I v I= Aw √(sin²wt + cos²wt)

= Aw ( constant )

acceleration

= dv / dt = - Aw² cos wt - Aw² sinwt

magnitude of acceleration

I a I = Aw²

= r w²

d ) centripetal force = m acceleration

m w² A  

=