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20) In the above drawing, M1 = 100 g, M2 = 110 g, M3 = 1.0 kg M4 = 115 g. M1 and M2 are free to slide without friction and the pulleys are massless. Find the accelerations of each block.

21) Repeat # 20, but this time the coefficient of kinetic friction between the table top and the blocks is 0.12.

22) Repeat # 21, however the pulleys are now wheels (I = (1/2)MR^2 ) of mass 75 g and radius 4 cm.

23)

A) How much work has been performed by gravity during the time needed for the system in #20 to move 1 m?

B) How much work has been performed by gravity during the time needed for the system in #22 to move 1 m?

24)

A) How much work has been performed by friction during the time needed for the system in #21 to move 1 m?

B) How much work has been performed by friction during the time needed for the system in #22 to move 1 m?

Explanation / Answer

M1 = 100 g, M2 = 110 g, M3 = 1.0 kg M4 = 115 g

(20)
Let the acceleration be a,
M3*g - T3 = M3 * a
1.0 * 9.8 - T3 = 1.0 * a --------1

T4 - M4*g = M4*a
T4 - 0.115 * 9.8 = 0.115 * a
T4 = 0.115 * 9.8 + 0.115 * a --------2

T3 - T4 = (M1+M2)*a
T3 - 0.115 * 9.8 - 0.115*a = (0.1 + 0.11)*a
9.8 - 1.0 * a - 0.115 * 9.8 - 0.115*a  = (0.1 + 0.11)*a
a = 6.55 m/s^2

21)
Let the acceleration be a,
M3*g - T3 = M3 * a
1.0 * 9.8 - T3 = 1.0 * a --------1

T4 - M4*g = M4*a
T4 - 0.115 * 9.8 = 0.115 * a --------2

T3 - T4 - uk*(M1+M2)*g = (M1+M2)*a
T3 - 0.115 * 9.8 - 0.115*a - 0.12 * 9.8 *(0.1 + 0.11) = (0.1 + 0.11)*a
9.8 - 1.0 * a - 0.115 * 9.8 - 0.115*a - 0.12 * 9.8 *(0.1 + 0.11)  = (0.1 + 0.11)*a
a = 6.36 m/s^2

(22)
T3 - T = I * (a)/r^2
T - T' - uk*(M1*g + M2*g) = (M1+M2)*a
T' - T4 =  I * (a)/r^2
T4 = M4*g + M4*a

Solving for a,

T' - (M4*g + M4*a) =  I * (a)/r^2
T' = I * (a)/r^2 + (M4*g + M4*a)

T - ( I * (a)/r^2 + (M4*g + M4*a) ) - uk*(M1*g + M2*g) = (M1+M2)*a
T = (M1+M2)*a + ( I * (a)/r^2 + (M4*g + M4*a) ) + uk*(M1*g + M2*g)

T3 - T = I * (a)/r^2
T3 - (M1+M2)*a - ( I * (a)/r^2 + (M4*g + M4*a) ) - uk*(M1*g + M2*g) = I * (a)/r^2
(1*9.8 - 1*a) - (0.1 + 0.11)*a - ( 1/2*0.075*a + (0.115*9.8 + 0.115*a) ) - 0.12*(0.1*9.8 + 0.11*9.8) = 1/2 * 0.075 * (a)
a = 6.01 m/s^2

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