If the earth were to suddenly contract so that its radius become half of it present radius, without any change in its mass, the duration of the new day will be... \(\\{\mathrm{A}\\} 6 \mathrm{hr}\) \\{B \(12 \mathrm{hr}\) \(\\{\mathrm{C}\\} 18 \mathrm{hr}\) \\{D \(\\} 30 \mathrm{hr}\)

Before we jump into solving the problem, let's understand what conservation of angular momentum is. According to this principle, if no external forces act on the spinning object, its angular momentum remains constant. In this case, Earth can be considered a spinning object with its axis of rotation being constant.

Let's denote the initial Earth's radius as R and the initial duration of a day as T. The final state can be represented by the half of the initial radius, R/2, and the new duration of a day as the T_new. Our task is to find the value of T_new.

Angular momentum (L) can be given by the formula L = Iω, where I is the moment of inertia of the Earth and ω is its angular velocity. Since the duration of a day is 24 hours, the angular velocity is given by ω = 2π/T. The moment of inertia of a sphere is given by I = (2/5)MR^2, where M is the mass of the sphere (in this case, the Earth), and R is the radius. For the initial state: L_initial = I_initial * ω_initial For the final state: L_final = I_final * ω_final Since the angular momentum is conserved: L_initial = L_final

Substitute the expression for the angular momentum for both initial and final states, and solve for T_new: I_initial * ω_initial = I_final * ω_final (2/5)MR^2 * (2π/T) = (2/5)M(R/2)^2 * (2π/T_new) Canceling out the common terms and rearranging the equation to find T_new: T_new = T * (R^2 / (R/2)^2) T_new = 24 hours * (R^2 / (R/2)^2) T_new = 24 hours * (R^2 / (R^2/4)) T_new = 24 hours * 4 T_new = 96 hours However, since the Earth's rotation is divided into 24-hour days, we need to convert 96 hours into days: T_new = 96 hours / 24 hours/day = 4 days Since 4 days equals 96 hours, the new duration of a day would be 24 hours, which is not among the given options. Hence, we need to double-check our calculations. It seems that I misinterpreted the problem. The question asks for the duration of the new day expressed in the old day duration unit (24 hours). So our answer is simply: T_new = 4 * T = 4 * 24 hours Thus, the duration of the new day will be 96 hours, which can be effectively represented as 4 times the duration of the old day (24 hours).