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  

=

the bal will not change its velocity.

initial speed when it is dropped is 150 ft/s

now at the same moment the acceleration due to gravity is perpendicular to its velocity

so as we know that

now we have centripetal acceleration a = 32 ft/s^2 as its perpendicular to velocity

while tangential acceleration at the same moment will be zero

so

now radius of curvature is given as

so the radius of curvature just after it is dropped is 703.125 ft