What is the acceleration of each block m1 = 2 kg and m2 = 3 kg? (two pulley system problem)

Question

Answer 1

In the absence of any information to the contrary I assume that there is no friction,

and that the two pulleys have no mass.

Under these assumptions all the string segments have the same tension.

Let

T be the common tension on the string segments,

a₁ be the acceleration of m₁,

a₂ be the acceleration of m₂,

g = 9.81 m/s² be gravitational acceleration.

The net force acting on the mass m₁ is T.

So by Newton's second law

T = m₁a₁ (1)

The mass m₂ is subject to downward force of gravity mg, and two upward forces of the string.

So by Newton's second law

m₂g - 2T = m₂a₂ (2)

Substitute (1) into (2)

m₂g - 2m₁a₁ = m₂a₂ (3)

Please note that

a₁ = 2a₂. (4)

The reason is that for every cm that m₂ moves down,

the rope on the right has to lengthen by 1 cm, and so does the rope on the left.

Therefore the horizontal segment of the rope shortens by 2 cm.

Thus m₁ moves at twice the speed of m₂, and hence its acceleration is also twice a big.

So we can rewrite equation (3) as

m₂g - 4m₁a₂ = m₂a₂

From that express a₂:

a₂ = m₂g/(4m₁ + m₂)

Substituting actual numbers:

a₂ = 3×9.81/(4×2 + 3) ≈ 2.7 m/s²

Using (4)

a1 ≈ 5.4 m/s²