Show below are two solid discs pulleys ( I =1/ 2 MR 2 ) connected to a hanging o

Question

Show below are two solid discs pulleys (I=1/2MR2) connected to a hanging object. The mass and radius of the two pulleys are dierent and are given in the gure. The mass of the hanging object is also given. Using energy considerations, calculate the speed of the hanging mass after falling a distance d in terms of the variables given. Hint: The change in potential energy of the hanging mass will result in the increase of three kinetic energy terms two rotational and one translational.

Please explain and show work! Thanks

Problem 3: Show below are two solid discs pulleys (1 = 1/2MR2) connected to hanging object. The mass and radius of the two pulleys are different and are given in the figure. The mass of the hanging object is also given. Using energy considerations, calculate the speed of the hanging mass after falling a distance d in terms of the variables given. Hint: The change in potential energy of the hanging mass will result in the increase of three kinetic energy terms two rotational and one translational. R, M 2R.AM 2 M

Explanation / Answer

let v is the speed of hanging block when it falls d distance.

Apply conservation of energy

gain in kinetic energy of three objects = loss of potentail energy of hanging object

0.5*I1*w1^2 + 0.5*I2*w2^2 + 0.5*(2M)*v^2 = (2*M)*g*d

0.5*(0.5*M*R1^2)*w1^2 + 0.5*(0.5*4*M*R2^2)*w2^2 + M*v^2 = 2*M*g*d (since V = R1*w1 = R2*w2)

0.25*M*v^2 + M*v^2 + M*v^2 = 2*M*g*d

2.25*M*v^2 = 2*M*g*d

2.25*v^2 = 2*g*d

v = sqrt(0.89*g*d) <<<<<<<-------------Answer

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