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.

(a) If a1 and a2 are the accelerations of m1 and m2, respectively, what is the relation between these accelerations? (Use a_1 for a1, m_1 for m1, and m_2 for m2 as appropriate.)
a2 =

(b) Find expressions for the tensions in the strings in terms of the masses m1 and m2, and g. (Use g, m_1 for m1, and m_2 for m2 as appropriate.)
T1 =
T2 =

(c) Find expressions for the accelerations a1 and a2 in terms of the masses m1 and m2, and g. (Use g, m_1 for m1, and m_2 for m2 as appropriate.)
a1 =
a2 =

Explanation / Answer

the acceleration is F/m
the only force action on the system is m1*g
and the masses that have to be accelerated are m1 and m2/2

a1 = m1g/(m1 + m2/2)
a2 = a1/2

a) a2 = a1/2

b) The tension T1 between m1 and second pulley is
T1 = m1g - m1a
T1 = m1g - m1(m1g/(m1 + m2/2)
T1 = m1g - m1^2g(m1 + m2/2)
Tension T2 between the second pulley and m2 is
T2 = T1/2

c) a1 = m1g/(m1 + m2/2)
a2 = a1/2

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