A water wheel with radius R_w = 12 m and mass M_w = 1.95 times 10^3 kg is used t

Question

A water wheel with radius R_w = 12 m and mass M_w = 1.95 times 10^3 kg is used to power a grain mill next to a river. Treat the water wheel as a hollow cylinder. The rushing water of the river rotates the wheel with a constant frequency f_r = 23 Hz. R_w = 1.2 m M_w = 1.95 times 10^3 kg f_r = 23 Hz Calculate the angular velocity omega_w of the water wheel in radians/sec. A numeric value is expected and not an expression. omega_w = ______ Calculate the kinetic energy K_w, in J, of the water wheel as it rotates. A numeric value is expected and not an expression. K_w = ______ Assume that every second, 20% of the kinetic energy of the water wheel is transmitted to the grain mill. Calculate the power P_w in W of the grain mill based on the energy it receives from the water wheel. A numeric value is expected and not an expression. P_w = ______

Explanation / Answer

A) angular velocity is W = 2*pi*f = 2*3.142*2.3 = 14.45 raad/sec

B) Kw = 0.5*I*w^2

I is the moment of inertia of wheel = M*R^2 = 1.95*10^3*(1.2^2) = 2808 Kg-m^2

w = 14.45 rad/sec

KW = 0.5*I*w^2 = 0.5*2808*14.45^2 = 2.93*10^5 J

C) Pw = 0.2*Kw/t = (0.2*2.93*10^5)/1 = 58600 W = 58.6 kW

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