An aeroplane is flying in circles of radius r_o at a speed of v_o giving the plane an angular velocity of w_o and a time period of T_o.

Question

What will the angular velocity w and time period T in terms of w_o and T_o when r=1.5r_o?

Answer 1

We can use the conservation of angular momentum to relate the angular velocity and time period of the airplane when it changes its radius of orbit from r_o to 1.5r_o.

The conservation of angular momentum states that the product of the moment of inertia (I) and the angular velocity (w) is conserved for a rotating body, provided that there are no external torques acting on it. For a point mass rotating in a circular orbit of radius r and velocity v, the moment of inertia is I = mr^2, where m is the mass of the object.

When the airplane changes its radius of orbit from r_o to 1.5r_o, the moment of inertia changes as (m(1.5r_o)^2)/(mr_o^2) = 2.25. Therefore, the angular velocity changes in the opposite direction to maintain the conservation of angular momentum. Mathematically, we can write:

Iw_oT_o = IwT

where I = m*r^2, w_o is the initial angular velocity, T_o is the initial time period, w is the new angular velocity, T is the new time period, and r is the new radius of orbit.

Substituting r = 1.5r_o and I = m*(1.5r_o)^2 = 2.25mr_o^2, we get:

w = w_o*(r_o/(1.5r_o))^2 = 0.44*w_o

T = T_o*(r_o/(1.5r_o))^2 = 1.94*T_o

Therefore, the angular velocity w is 0.44 times the initial angular velocity w_o, and the time period T is 1.94 times the initial time period T_o.