Inclined Plane Force Calculator
Slope push force, normal force, and grade from mass.
What it does
Slopes split weight into two directions. Part pulls the load downhill along the plane, part presses it into the surface, and steeper angles shift share from pressing to pulling. A 100 kg crate on 30 degrees needs 490.33 N held back while 849.28 N bears on the boards.
The tool above resolves both parts in either unit system. Enter mass and angle for newtons and grade, or pounds for lbf directly. Metric defaults show the 100 kg crate; imperial mirrors at 220 lb needing 110 lbf up the slope.
University treatments match term for term. MIT classical mechanics works sliding blocks on inclines, RMIT derives mg sin against mg cos with a worked toy car, and Georgia Tech prints the same components with a 2.5 kg example. OpenStax and LibreTexts carry identical friction extensions.
Grade speaks the same geometry to drivers. Tangent times 100 turns 30 degrees into 57.74 % and 45 into 100 %, the numbers on warning signs. Friction stays outside this page: rough planes add mu times normal to every push.
How to use
- Pick metric (kg) or imperial (lb).
- Enter the mass or weight.
- Enter the slope angle in degrees.
- Read push force first, then normal.
- Read grade percent for signage.
- Add friction separately on rough planes.
Formula
- mass
- Sliding mass in kg, or weight in lb.
- angle
- Slope angle in degrees, 0 to 90 exclusive.
- 9.80665
- Standard gravity, m/s2.
Frictionless static case from weight resolution (MIT, RMIT, Georgia Tech). Imperial pounds read as force directly.
Worked example
Example: 100 kg at 30 degrees
Metric defaults: 100 kg at 30 degrees. Imperial mirrors at 220 lb.
- Inputs: 100 kg mass, 30 degree slope.
- Sine: sin(30) = 0.5. Push: 100 x 9.80665 x 0.5 = 490.33 N.
- Cosine: cos(30) = 0.86603. Normal: 980.665 x 0.86603 = 849.28 N.
- Grade: tan(30) x 100 = 57.74 %.
- Tool shows 490.33 N, 849.28 N, 57.74% grade.
Reference table
| Angle | Push (N) | Normal (N) | Grade |
|---|---|---|---|
| 5 deg | 85.47 | 976.93 | 8.75 % |
| 15 deg | 253.82 | 947.25 | 26.79 % |
| 30 deg | 490.33 | 849.28 | 57.74 % |
| 45 deg (200 kg) | 1386.87 | 1386.87 | 100 % |
Limits
- Frictionless only. Rough planes add mu x N.
- Static holds; accelerating loads differ.
- Point mass assumed; rolling changes answers.
- Grade undefined at vertical.
Common mistakes
- Mixing sin and cos between push and normal.
- Entering grade percent as the angle.
- Using pounds mass where force belongs.
- Forgetting friction on real ramps.
- Reading 45 degrees as 45 % grade.
Try it yourself
Quick checks. Answers sit tucked under each question.
50 kg at 20 degrees?
sin20 = 0.34202. F = 490.3325 x 0.34202 = 167.7 N. N = 460.76 N. Grade 36.4 %.
200 kg at 45 degrees?
sin45 = cos45 = 0.70711. F = N = 1961.33 x 0.70711 = 1386.87 N. Grade 100 %.
Questions
How is push force figured?
Mass times g times sin(angle). Here 100 x 9.80665 x 0.5 = 490.33 N.
How is normal force figured?
Mass times g times cos(angle). Here 849.28 N presses into the plane.
What is grade percent?
Tangent times 100. A 30 degree slope is 57.74 %; 45 degrees is 100 %.
Does imperial work directly?
Yes. Imperial pounds are force units: 220 lb at 30 degrees needs 110 lbf up the slope.
What about friction?
Add mu times normal to the push. Rough planes need friction math beyond this page.
What happens at 90 degrees?
Yes: sin(90) = 1, so the full weight hangs. Grade math breaks at vertical (tan 90 undefined).
Sources
- MIT OCW: Sliding Block on Incline (8.01SC)
- RMIT Learning Lab: Forces on Slopes (mg sin/cos)
- Georgia Tech Physics: Inclined Plane