An object of mass m1 hangs from a string that passes over a very light fixed pul

Question

An object of mass m1 hangs from a string that passes over a very light fixed pulley P1 as shown in the figure below. The string connects to a second very light pulley P2. A second string passes around this pulley with one end attached to a wall and the other to an object of mass m2 on a frictionless, horizontal table. P2 ing mi (a) If a1 and a2 are the accelerations of m, and m2, respectively, what is the relation between these accelerations? (Use any variable or symbol stated above as necessary.) (b) Find expressions for the tensions in the strings in terms of the masses m1 and m2, and g " c) Find expressions for the accelerations a1 and a2 in terms of the masses m and m2, and . 31

Explanation / Answer

given, objects of mass m1 and m2
light pulleys P1 and P2, massless strings, frictionless table

a. from the given diagram
   accelerations of mass m1 and m2 = a1 and a2 respectively
   now, for every unit distance moved by mass m1, m2 will move half a unit
   hence a1 = 2a2
   or a2 = a1/2

b. tension in string connected to masses m1 and m2 = T1 and T2 respectively
   now from force balance on m1
   m1g - T1 = m1a1
   and force balance on m2
   m2a2 = T2
   also, force balance at pulley P2
   2T2 = T1

   hence
   m1g - 2(m2a2) = m1*2a2
   m1g = a2(2m1 + 2m2)
   hence
   T1 = m1g - m1a1 = m1g - m1*2a2
   T1 = m1g - m1*m1g/(m1 + m2) = m1g(m1 + m2 - m1)/(m1 + m2)
   T1 = m1*m2*g/(m1 + m2)

   T2 = T1/2 = m1*m2*g/2(m1 + m2)

c. a2 = m1*g/2(m1 + m2)
   a1 = 2a2 = m1*g/(m1 + m2)

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