This inclined plane calculator is used to analyze the forces and motion of objects on sloped surfaces using F = W sin(θ) + μ W * cos(θ) formula.
For example, a box of mass (m = 10 kg) is placed on an inclined plane at an angle (θ = 30°). Using the inclined plane calculator, one can input the mass and angle to find:
Force perpendicular to the incline: F_perpendicular = mg cos(θ) = 10 9.81 * cos(30°) = 84.87 N
Force parallel to the incline: F_parallel = mg sin(θ) = 10 9.81 * sin(30°) = 49.05 N
Inclined Plane Calculator
| Mass (kg) | Angle (°) | Friction Coefficient | Force Required (N) | Acceleration (m/s²) |
|---|---|---|---|---|
| 10 | 15 | 0.05 | 43.2 | 2.01 |
| 25 | 30 | 0.2 | 164.9 | 3.42 |
| 50 | 45 | 0.1 | 417.3 | 5.89 |
| 100 | 20 | 0.15 | 451.6 | 2.57 |
| 75 | 35 | 0.25 | 534.8 | 3.76 |
| 15 | 10 | 0.1 | 25.5 | 1.17 |
| 40 | 25 | 0.3 | 214.5 | 4.12 |
| 60 | 40 | 0.2 | 471.5 | 6.25 |
| 80 | 50 | 0.15 | 682.2 | 4.84 |
| 120 | 15 | 0.1 | 295.4 | 3.09 |
| 90 | 30 | 0.25 | 432.7 | 5.01 |
| 30 | 45 | 0.2 | 245.0 | 4.10 |
| 55 | 20 | 0.05 | 207.6 | 2.89 |
| 70 | 35 | 0.3 | 556.3 | 4.78 |
| 110 | 10 | 0.2 | 194.5 | 2.45 |
| 125 | 25 | 0.15 | 676.7 | 5.67 |
Inclined Plane Formula
F = W * sin(θ)This formula assumes ideal conditions with no friction. In real-world scenarios, we must consider friction, leading to a more comprehensive formula:
F = W sin(θ) + μ W * cos(θ)Where:
- μ is the coefficient of friction
- W * cos(θ) represents the normal force
A 100 kg crate on a 30° ramp with a friction coefficient of 0.2.
Weight (W) = 100 kg 9.8 m/s² = *980 N
F = 980 sin(30°) + 0.2 980 * cos(30°)
F = 490 + 169.7 = 659.7 N
Force Parallel to the Incline (F_parallel):
F_parallel = mg sin(theta)Where:
- m = mass (kg)
- g = gravitational acceleration (9.81 m/s²)
- theta = angle of the incline (degrees)
Normal Force (F_normal):
F_normal = mg cos(theta)Frictional Force (F_friction, if there’s friction):
F_friction = μ F_normal = μ mg cos(theta)Where:
- μ = coefficient of friction
Net Force (if friction is present):
F_net = F_parallel - F_frictionFormula for Inclined Plane with Pulley
F = (W sin(θ) + μ W * cos(θ)) / nWhere n is the number of supporting rope strands in the pulley system.
With a double pulley system (n=2): F = 659.7 / 2 = 329.85 N
When a pulley system is incorporated with an inclined plane, it can significantly reduce the force required to move an object. The basic formula remains similar, but the mechanical advantage of the pulley system is factored in:
How to Calculate an Inclined Plane
- Identify given information: Mass, angle, coefficient of friction, etc.
- Convert units if necessary (e.g., kg to N for weight)
- Draw a free body diagram showing all forces
- Apply the appropriate formula based on the scenario
- Solve for the unknown variable
Problem: Calculate the acceleration of a 5 kg block sliding down a 25° incline with a friction coefficient of 0.1.
Step 1: Identify given information
- Mass (m) = 5 kg
- Angle (θ) = 25°
- Coefficient of friction (μ) = 0.1
- Acceleration due to gravity (g) = 9.8 m/s²
Step 2: No unit conversion needed
Step 3: Draw a free body diagram (forces: weight, normal force, friction)
Step 4: Apply Newton’s Second Law along the incline F = ma
mg sin(θ) – μ mg * cos(θ) = ma
Step 5: Solve for acceleration (a)
a = g (sin(θ) – μ cos(θ))
a = 9.8 (sin(25°) – 0.1 cos(25°))
a = 9.8 (0.4226 – 0.1 0.9063)
a = 3.24 m/s²
The block accelerates down the incline at 3.24 m/s².
Related Tools:
