Question

Problem: Learning to fly

In this example, we will apply conservation of energy, including potential energy, together with Newton’s laws and the expression for the acceleration in the radial direction of an object moving on a circular path.

A young fledgling bird of mass m is sat at the very top of a dome, which is circular in cross section of radius R. Starting with an initial velocity of zero, the bird starts to slide down the dome. There is zero friction between the surface of the dome and the young bird’s tummy. The goal of this problem is to determine when the young bird takes off.

a) Sketch the situation.

b) Draw a force diagram that shows all forces on the baby bird, when the bird has slid, so that its location on the dome make an angle θ with the vertical through the top of the dome.

c) What is the component of the net force in the radial direction?

d) What is the relationship between angular velocity and velocity (speed) for motion in a circle?

e) Given that the radial component of the acceleration of a particle moving in a circle of radius R with speed v is −v 2/R write down Newton’s Second Law for the baby bird in terms of m, g, v, R, θ, and the normal force (N) exerted by the dome on the baby bird.

f) Using conservation of energy, write down an equation linking v, R, g, and θ.

g) What is the value of the normal force at the moment when the baby bird just looses contact with the dome?

h) How far from the top does the baby bird lose contact with the dome?

Answer 1