An object of mass m 1 hangs from a string that passes over a very light fixed pu

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 any variable or symbol stated above as necessary.)
a2 =

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

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

Explanation / Answer

(a)

if m2 moves x far, so m1 move 2x far, finally

a1 = 2 (a2)...............................................(1)

a2=a1/2


(b).

free body of m2

F vertical = ma

m2 g - T2 = m2 a2

a2 = (m2 g - T2)/m2...................(2)

freebody of pulley P1

F horizontal = ma

T2 - 2T1 = m_pulley 1 (a2) = 0 ......remember m_pulley = 0

T2 = 2T1.................................(3)

free body of m1

consider for eqs (1), eqs (2) and eqs (3)

F horizontal = ma

T1 = m1 a1 = 2m1 a2

T1 = 2m1 (m2 g - T2)/m2

T1 = 2m1 (m2 g - 2T1)/m2

T1 = 2m1 m2 g/(4 m1 + m2)

T2 = 2T1

T2 = 2 (2m1 m2 g/(4 m1 + m2))

T2 = 4m1 m2 g/(4 m1 + m2)

(c).

remind of eqs (2)

a2 = (m2 g - T2)/m2

a2 = (m2 g - (4m1 m2 g/(4 m1 + m2)))/m2

a2 = g (1 - 4m1/(4m1 + m2))

a2 = g(m2/(4m1 + m2))

a1 = 2 (a2) = 2 g(m2/(4m1 + m2))

a1 = 2 g(m2/(4m1 + m2))

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