Inclined Plane Calculator: Sliding and Rolling Down a Ramp, Friction, Acceleration and Force Needed
Everything about an object on a ramp, with the working.
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How this calculator works
Choose a sliding block or a rolling shape.
Enter the mass, the angle, the friction coefficients and the length of the slope.
The calculator gives the forces, the acceleration, the time and the speed, and the forces needed to push the block up.
Formulas used
- a = g(\sin\theta - \mu_k\cos\theta)
- a = \frac{g\sin\theta}{1 + I/(mr^2)}
- \theta_s = \arctan\mu_s
- F = mg(\sin\theta + \mu\cos\theta)
Worked example
With the default numbers in the calculator above:
- Component of the weight along the slope (down the slope): 49.033 N
- Normal force from the slope (component of the weight into the slope): 84.928 N
- Largest static friction force: 42.464 N
- The angle at which a block just starts to slide (the angle of repose): 26.565 degrees
- Acceleration down the slope once it slides: 2.3555 m/s2
- Time to slide the length of the slope from rest: 2.0604 s
- Speed at the bottom: 4.8533 m/s (17.472 km/h)
- Force along the slope to start the block moving up: 91.497 N
- Force along the slope to push it up at a steady speed: 74.512 N
- Work done raising it up the slope at steady speed, and the part that is lost to friction: 372.56 J, of which 127.39 J is friction
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: What is on the slope? A solid cylinder or disc rolling.
- Component of the weight along the slope (down the slope): 49.033 N
- Normal force from the slope (component of the weight into the slope): 84.928 N
- Largest static friction force: 42.464 N
- Shape factor k = I / (m r squared), the share of the energy that goes into rotation: 0.5
- Acceleration down the slope when it rolls without slipping: 3.2689 m/s2 (a sliding block with no friction would have 4.9033)
- Time to roll the length of the slope from rest: 1.749 s
- Speed of its centre at the bottom: 5.7174 m/s
- Friction force needed to stop it slipping: 16.344 N
- Smallest static friction coefficient for pure rolling at this angle: 0.1925
Common questions
When does a block start to slide down a slope?
When the angle is more than the inverse tangent of the static friction coefficient.
Which rolls down fastest, a sphere, a cylinder or a hoop?
The solid sphere is fastest, then the solid cylinder, then the hoop, whatever their mass or size.
Does the mass change the acceleration?
No. The mass cancels, so heavy and light objects accelerate the same.
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Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any first-year physics text (inclined planes and rolling motion).
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