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physics.fisikastudycenter.com - How to find the ratio of strings tensions on a couple of blocks travel at frictionless surface. Given 5 blocks being pulled by a force F = 30 N along x direction. The masses are m₁ = 1 kg, m₂ = 2 kg, m₃ = 3 kg, m₄ = 4 kg, m₅ = 5 kg. You have to find the ratio of T₁ and T₂ . See the figure below and the following steps both normally and the short-cut.

Step 1 : Find the acceleration of the system (5 blocks), the force F divided by the sum of all block masses, it's a simple step even you don't need to use your pen to finish this. We have the system acceleration a = 30 : (1 + 2 + 3 + 4 + 5) = 2 m/s².

Step 2 : Find the tensions T₁ and T₂, use normal methods.
T₁
Σ F = ma
F−T₁ = m₁a
30− T₁ = 1(2)
T₁ = 28 N

T₂
T₁−T₂ = m₂a
28−T2 = 2(2)
T2 = 24 N

The ratio
T₁ : T₂ = 28 : 24
T₁ : T₂ = 7 : 6

Short-cut

Now, quickly try to find the ratio of :
1) T₁ : T₃
2) T₃ : T₄