The rate of growth of the mass M of a spherical raindrop fallingthrough a partic

Question

The rate of growth of the mass M of a spherical raindrop fallingthrough a particular cloud is given by M = rho, 4/3 Pir^3 and C is a constant. A.) Eliminate M from the above equation, so that the size of thedrop is expressed solely in terms of the radius r. B.) Separate variables and integrate to find an expression forr(t), given an initial radius r_o at time t = 0 c.)If r_o = 0.1 mm and C = 30 kg/ (sec m^3 ),find the radius after the drop has grown for 10 min. Answer: 0.42mm Hint: You don't need to know anything about the physics of raindropgrowth to solve this problem.dM/dt = ~ C r^3 Where

Explanation / Answer

We have that M = 4r3/3. Taking thederivative of this with respect to time gives that dM/dt =4r2(dr/dt) = Cr3. This isthe same as (4/r)dr = Cdt. Integrating both sidesgives that 4ln(r) + Co= Ct. Knowingthat at t = 0, r = ro, it follows that Co =-rln(ro), so the equation is the same as4ln(r/ro) = Ct. This is the same asln(r/ro) = Ct/(4), so we have that r =roeCt/(4). Plugging in t =600s, ro = 0.1 mm, C = 30kg / (sec m3), and = 1000 kg/m3 gives that r = 0.42mm.

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