Ladder Equilibrium Calculator: Wall and Floor Reactions and Friction Needed

Reactions and friction for a ladder leaning on a smooth wall, with a person standing on it.

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

Enter the ladder length and angle, its mass, the person's mass and how far up they are, and the floor friction. The calculator balances the vertical forces to get the floor reaction, takes moments about the foot to get the wall reaction, and sets the friction at the foot equal to the wall reaction.

It then compares the friction needed with the floor can give, gives the furthest the person can climb before the foot slips, and the smallest angle at which the ladder holds with the person at your chosen position.

This is a statics model of a smooth wall. For work on a real ladder follow the workplace rules, tie the ladder off or have it held at the foot, and set it at about 1 out for every 4 up.

Formulas used

  • N_f = (m_L + m_p)g
  • N_w = \frac{g\cos\theta\,(m_L L/2 + m_p x)}{L\sin\theta}
  • \mu_{needed} = \frac{N_w}{N_f}

Worked example

With the default numbers in the calculator above:

  1. Floor reaction (vertical forces balance): 882.90 N
  2. Wall reaction (moments about the foot of the ladder): 139.31 N
  3. Friction needed at the foot (horizontal forces balance): 139.31 N
  4. Friction coefficient needed: 0.1578
  5. Furthest the person can climb before the foot slips: the whole length (8.08 m calculated)
  6. Smallest safe angle with the person at this position: 55.81 °

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: Angle of the ladder from the floor 60 °; Distance of the person up the ladder 5 m.

  1. Floor reaction (vertical forces balance): 882.90 N
  2. Wall reaction (moments about the foot of the ladder): 481.42 N
  3. Friction needed at the foot (horizontal forces balance): 481.42 N
  4. Friction coefficient needed: 0.5453
  5. Furthest the person can climb before the foot slips: 3.58 m
  6. Smallest safe angle with the person at this position: 67.05 °

Common questions

Why does a steeper ladder hold better?

A steeper ladder pushes less sideways on the wall, so the foot needs less friction to stay put.

Does the person's position matter?

Yes. The higher the person goes, the bigger the sideways push at the wall and the more friction needed at the foot.

Is this the safe set-up for work?

No. It is a physics model. Use the ladder set-up calculator and your workplace rules for safe use.

Related calculators

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

Further reading: Meriam and Kraige, Engineering Mechanics: Statics (equilibrium with friction); AS/NZS 1892 (portable ladders).

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