Friction Calculator: Angle of Repose, Stopping Distance and Block Problems

Solve the common friction problems: when a block slips on a slope, how far a body slides before it stops, when stacked blocks slip, and the smallest pull that moves a block.

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

Friction force is the coefficient of friction times the normal force. A block on a slope slips when the tangent of the slope angle is larger than the static coefficient, which is why the angle of repose has tangent equal to the coefficient.

A sliding body is slowed by the friction force at a deceleration of the coefficient times g. The stopping distance is the speed squared divided by twice that deceleration, so doubling the speed makes the distance four times longer.

For the least force to drag a block the pull is angled upward. This lifts some weight off the floor, and the best angle has tangent equal to the coefficient.

Formulas used

  • F_f = \mu N
  • s = \dfrac{v^2}{2\mu g}
  • F_{max} = \mu_s(m + M)g
  • F_{min} = \dfrac{\mu m g}{\sqrt{1+\mu^2}}

Worked example

With the default numbers in the calculator above:

  1. Angle of repose (tilt at which it just slips): 26.57 °
  2. Force pulling the block down the slope: 9.81 N
  3. Largest friction that can hold it: 8.50 N
  4. Acceleration down the slope once it slips: 1.51 m/s²
  5. Time to slide the length of the slope from rest: 1.63 s
  6. Speed at the bottom: 2.45 m/s

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 problem? Friction coefficient from a rough slope against a smooth slope (a race); Angle of the slope from the horizontal 45 °.

  1. Friction coefficient of the rough slope: 0.36
  2. Acceleration down the smooth slope: 6.94 m/s²
  3. Acceleration down the rough slope: 4.44 m/s²
  4. Time down the smooth slope and the rough slope: 0.76 s and 0.95 s

Common questions

Does the mass change the stopping distance?

No. Friction and inertia both go with the mass, so a heavy and a light body slide the same distance from the same speed on the same surface.

What is the angle of repose?

The steepest slope on which a body stays still. Its tangent equals the static friction coefficient.

Why 9.81?

It is the acceleration of gravity in m/s². Many textbook sheets use 10 for easy sums, which gives slightly larger answers.

Related calculators

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

Further reading: Standard physics and engineering mechanics texts (friction and motion on slopes).

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