Strings and Pulleys Calculator: Tension, Atwood Machine and Connected Masses

Find the tension and the acceleration for the common string and pulley cases: hanging masses, strings at angles, the Atwood machine, connected masses on slopes and a movable pulley.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

A hanging mass that speeds up upward needs a string pull of m(g + a). Two strings holding a mass at angles share the weight: the vertical parts of the pulls carry it, and the sideways parts balance each other.

In an Atwood machine the heavier mass goes down with an acceleration of the difference in the masses times g over the total mass, and the tension is twice the product of the masses times g over the total.

For masses connected over a pulley the net pull along the string, less the friction, accelerates both masses together. A movable pulley halves the pull but doubles the rope pulled in.

Formulas used

  • T = m(g + a)
  • a = \dfrac{(m_2 - m_1)g}{m_1 + m_2}
  • T = \dfrac{2 m_1 m_2 g}{m_1 + m_2}
  • 2T - Mg = Ma

Worked example

With the default numbers in the calculator above:

  1. Acceleration of the masses: 1.96 m/s² (the heavier mass goes down)
  2. Tension in the string: 23.54 N
  3. Total pull on the pulley support: 47.09 N

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Which case? Hanging mass, at rest, speeding up or slowing down; Mass 1 (the single hanging mass, the load, or the mass on the table or slope) 80 kg; Upward acceleration (hanging mass and movable pulley; negative is downward) 4 m/s².

  1. Weight of the mass: 784.80 N
  2. Tension in the string: 1104.80 N

Common questions

Why is the tension in an Atwood machine less than the heavier weight?

Because the heavier mass is accelerating downward. The string supports only part of its weight.

What if the strings are very flat?

The tension grows without limit as the angle to the horizontal goes to zero, because the string must supply the whole vertical support with a very small vertical part.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Standard physics texts (Newton second law, strings and pulleys). Lifting must be done under the lifting rules for your site.

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