Two rough planes, A and B, inclined at 30 and 60 to the horizontal, respectively

Question

Two rough planes, A and B, inclined at 30 and 60 to the horizontal, respectively, and at the same vertical height, are placed back to back. A pulley of mass 1 kg and radius 10 cm is fixed to the top of the system. A string is passed over the pulley and attached to two masses, mA = 2 kg and mB, = 6 kg. The coefficient of kinetic friction between the masses and the planes is 0.2.

a. When released from rest, what is the acceleration of the system?

b. Find the tension in the string on side A.

c.Find the tension in the string on side B.

d.If the peak of the system is 1.1 m vertically above the ground and block B is at the top of the incline when released from rest, how much time will it take for block B to reach the bottom? !!!!!!!!!!!!!

e. What is the work done by friction on the entire system in this time? !!!!!!!!!!!!!!!!

PLEASE

Explanation / Answer

I ( of pulley) = 1/2 ( 1) ( 0.01) = 0.005 kg m^2

Let Ta and Tb be the tension in ropes,

equation of motion for A: 2 (a) = TA - 2(g) sin 30 - 0.2 ( 2) (g) cos 30 --------eq(1)

2a= Ta- 9.8 - 3.94

2a= Ta- 13.194 -----eq(2)

Equation of motion for B: 6 (a) = 6 (g) sin 60 - TB - 0.2 (6) gcos 60 ----------eq(3)

6a= 50.922- Tb - 5.88

6a= 45.042 - Tb-------eq(4)

addiing eq(2) and eq (4),

8a= Ta- Tb + 31.848----eq(5)

= r ( Tb-Ta) = I ( a/r) = 1/2 m a

Tb- Ta = ( 1/2r) ma

Tb- Ta = 0.5 a/ 0.1= 5a-----eq(6)

combining eq(5) and eq(6)

8a=- 5a +31.848

13a= 31.848

a( acceleration) = 2.45m/s^2 apprx

b)2a= Ta- 5.86

Ta= 10.759.N

c)  6a= 45.042 - Tb

tb = 45.042 - 6 (2.45) = 30.342 N

d) slant length of block B displacement = 1.1 /sin 60 =1.27 m

1,27 = 1/2 ( 2.45t^2)

t = 1.02 sec appprx

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