Assume: The positive y direction is up. A pulley (in the form of a uniform disk)

Question

Assume: The positive y direction is up.
A pulley (in the form of a uniform disk)
withmass 65 kg and a radius 11 cmis attached
to the ceiling, in a uniform gravitational field,
and rotates with no friction about its pivot.
The acceleration of gravity is 9.8 m/s2 .
These masses are connected by a massless
inextensible cord. T1, T2, and T3 are magnitudes
of the tensions.

a)Determine the acceleration of the mass
23 kg.

b)Determine the acceleration of the center of
mass ~acm of M2 +m1.

c)Determine the magnitude of the tension T1.

d)Determine the magnitude of the difference
in the tensions T = T2 ? T1.

e)Determine the magnitude of the tension T3.

f)Determine the magnitude of the difference
in the force of gravity on the sum of the three
masses and the tension in the cord holding up
the pulley Fg = (Mp +M2 + m1) g ? T3.

Explanation / Answer

a)

Writing equations of motions :

for M1 = 23 kg,

T1 - M1*g = M1*a -------- (1)

for M2 :

M2*g - T2 = M2*a -----------(2)

Adding the two equations,

T1 - T2 + (M2-M1)*g = (M1+M2)*a -------- (3)

For the pulley:

(T2 - T1)*R = I*A

where I = 0.5*MR^2

A = angular acceleration = a/R

So, (T2 - T1)*R = 0.5*MR^2*(a/R)

So, T2- T1 = 0.5*M*a ----------- (4)

Adding equations (3) and (4), we get:

(M2-M1)*g = 0.5*M*a + (M1+M2)*a

So, a = (M2-M1)*g/(0.5*M + (M1+M2))

So, a = (44-23)*9.8/(0.5*65+(44+23))

So, a = 2.07 m/s2 <---------answer(acceleration)

b)

acceleration of center of mass = (M1*a - M2*a)/(M1+M2) = 0.649 m/s2

c)

T1 = M1*(g+a) = 23*(9.8+2.07) = 273 N

d)

T2 - T1 = 0.5*M*a = 0.5*65*2.07 = 67.3 N <-----------answer

e)

T3 = T2 - T1 = 67.3 N <-------answer

f)

Fg = (M+M2 + M1) g - T3.= (65+44+23)*9.8 - 67.3 = 1226.3 N <--------answer

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