A wagon wheel is made of an outer rim and several rod shaped spokes. The rim is

Question

A wagon wheel is made of an outer rim and several rod shaped spokes. The rim is a ring with a mass of 5.20 kg and a radius of 0.600m.Attached to the wheel are                     12 rods attached to the rim of the wheel and the center, each with a length equal to the radius of the wheel and a mass of 0.450 kg. The wheel rolls without                     slipping on a horizontal upper level floor as shown in the figure. The wheel rolls to the right with an initial velocity of 0.550m/s, 1.00 m above the lower                     level.(A)Set up the integral for calculating the moment of inertial for the rim of the wheel.You do not need to solve it.(B)The wheel rolls down a ramp.What is                     the wheel's speed on the lower level.

Explanation / Answer

a)
let m1 + m2 = M = 5.65 kg

I_ring = m1*R^2

I_rod = m2*R^2/3


total moment of inertia ofsystem,

I = I_ring + 12*I_rod

= m1*R^2 + 12*m2*R^2/3

= (3*m1*R^2+12*m2*R^2)/3

= 5*(m1+m2)*R^2

= 5*M*R^2

b)

accrdoing to conservation of enrgy

0.5*M*v1^2 + 0.5*I*w1^2 + m*g*h = 0.5*M*v2^2 + 0.5*I*w2^2

0.5*M*v1^2 + 0.5*5*M*R^2*w1^2 + m*g*h = 0.5*M*v2^2 + 0.5*5*M*R^2*w2^2


0.5*M*v1^2 + 2.5*M*v1^2 +m*g*h = 0.5*M*v2^2 + 2.5*M*v2^2


3*M*v1^2 + M*g*h = 3*M*v2^2

v2^2 = v1^2 + g*h/3

v2 = sqrt(v1^2 + g*h/3)

= sqrt(0.55^2 + 9.8*1/3)

= 1.889 m/s

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