Find out the friction force, Mechanical Engineering

Find out the friction force:

The cord passes over a massless and frictionless pulley, carrying a mass M1 at one end and wrapped around a cylinder of mass M2 which rolls on a horizontal plane. What is the acceleration of mass M1?

Solution

 We should keep in mind that acceleration a1 of mass M1 is not equal to the acceleration a of the mass centre of the cylinder.

Considering dynamic equilibrium, for cylinder we may write

∑ Fx  = 0,            ∴ T - M 2  a¯ - F = 0  -------- (1)

∑ M o  = 0,            T r + F . r - I α = 0  -------- (2)

and for mass M1, we have

∑ Fy  = 0,     T - M1  g + M1 a1 = 0           -------- (3)

and, we have,

a¯ = r α

and, a1  = 2 r α

i.e., a1  = 2 a¯

and

I =( ½) M2 r 2 , a¯ = r α

∴  From Eq. (1) above, we obtain

T - F = M 2  a¯

From Eq. (2)

 (T + F) r = (½) M 2 r 2 α

T + F = (1/2)M 2  r α

T  + F  = 1  M2

∴ 2T = M 2  a¯ + (1 /2)M2   a

∴  T  = (¾) M2 a¯                       ---------- (4)

Figure 7.17

From Eq. (3), we get

 T = M 1  g - M1  a1

∴ From Eq. (4),

T = (¾) M2a=M1g-M1a1

But a¯ = (1/2)a1

∴ 3/8 M2a1 = M1 g - M1 a1

∴ a1((3/8)  M 2  + M1 ) = M1  g

∴  a1  =  M1 g/(3/8) M 2  + M1 )

Posted Date: 1/29/2013 6:43:56 AM | Location : United States







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