Circular Motion Calculator: Banked Curve, Level Turn, Conical Pendulum and Vertical Circle

Level and banked turns, a conical pendulum, and a mass swung in a vertical circle.

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

Choose the case and enter the mass, radius and speed. For a car on a level bend the friction supplies the inward force, so the calculator gives the force needed, the friction coefficient needed, and the greatest speed before the car skids.

For a banked curve it gives the design speed at which no friction is needed, and the least and greatest speeds the friction can hold. For a conical pendulum it gives the tension, the angular speed, the speed and the time for one turn. For a vertical circle it gives the tension at the bottom and the top, and the least speed needed to complete the circle.

Use it for road and track design, rotating equipment and physics problems. The friction coefficient should be the wet-road value for design.

Formulas used

  • F = \frac{mv^2}{r}
  • v = \sqrt{rg\tan\theta}
  • \omega = \sqrt{\frac{g}{L\cos\theta}}

Worked example

With the default numbers in the calculator above:

  1. Inward force needed to turn (supplied by friction): 4500.00 N
  2. Friction coefficient needed at this speed: 0.4587
  3. Greatest speed before it skids: 18.53 m/s (66.71 km/h)

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? Banked curve.

  1. Speed at which no friction is needed (the design speed): 9.30 m/s (33.48 km/h)
  2. Greatest speed before sliding up the bank: 22.14 m/s (79.72 km/h)
  3. Least speed before sliding down the bank: 0: friction alone holds a stopped car on this bank

Common questions

What provides the force to turn a car?

Friction between the tyres and the road on a level bend. On a banked curve, part of the normal force from the road also points to the centre.

Why does a faster car skid?

The inward force needed rises with the square of the speed, so doubling the speed needs four times the friction.

What speed must a mass have to swing over the top?

At the top the string stays taut if the speed there is at least the square root of g times r, which needs the square root of 5 g r at the bottom.

Related calculators

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

Further reading: Halliday, Resnick and Walker, Fundamentals of Physics (uniform circular motion); Meriam and Kraige, Engineering Mechanics: Dynamics.

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