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 51 kg and a radius 18 cm is 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.

Q1)

Determine the acceleration of the mass
21 kg.
Answer in units of m/s^2.

Q2)

Determine the acceleration of the center of
mass acm of M2 +m1.
Answer in units of m/s^2.

Q3)

Determine the magnitude of the tension T1.
Answer in units of N.

Q4)

Determine the magnitude of the difference
in the tensions DELTAT = T2 ? T1.
Answer in units of N.

Q5)

Determine the magnitude of the tension T3.

Answer in units of Newtons.

Q6)

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 DELTAFg = (Mp +M2 +m1) g ? T3.
Answer in units of N.

Help on any one of these parts would be greatly appreciated. Feel free to answer in sections. I have tried numerous parts and am having difficulty grasping this concept.

Assume: The positive y direction is up. A pulley (in the form of a uniform disk) withmass 51 kg and a radius 18 cm is 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. Q1) Determine the acceleration of the mass 21 kg. Answer in units of m/s^2. Q2) Determine the acceleration of the center of mass acm of M2 +m1. Answer in units of m/s^2. Q3) Determine the magnitude of the tension T1. Answer in units of N. Q4) Determine the magnitude of the difference in the tensions DELTAT = T2 ? T1. Answer in units of N. Q5) Determine the magnitude of the tension T3. Answer in units of Newtons. Q6) 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 DELTAFg = (Mp +M2 +m1) g ? T3. Answer in units of N. Help on any one of these parts would be greatly appreciated. Feel free to answer in sections. I have tried numerous parts and am having difficulty grasping this concept.

Explanation / Answer

By free-body analysis, we get the following equations:

36*g - T2 = 36*a

T1 - 21*g = 21*a

Torque = (T2 - T1)*0.18 = I = (51*0.182/2)*

But angular acceleration = a/r

So, Torque = (T2 - T1)*0.18 = I = (51*0.182/2)*a/0.18

Putting T1 and T2 from first two equations into this equation we get,

[36(g-a) - 21(g+a)]*0.18 = (51*0.182/2)*a/0.18

(a) Solving this gives a = 1.784 m/s2

(b) Since both the masses are moving with same acceleration in opposite directions, the acceleration of the centre of combined mass will be same.

(c) and (d) T1 and T2 can be found using above equation.s.

(e) T3 = T1 + T2 + 51*g

(f) (51 + 36 + 21)*g - T3 can be found from above.

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