Forces on an Inclined Plane

When an object rests on an inclined surface, its weight (mg) can be split into two components: one perpendicular to the surface (mg·cos θ) and one parallel to the surface (mg·sin θ). The perpendicular component is balanced by the normal force from the surface. The parallel component tries to pull the object down the slope and must be countered by friction (and any applied forces) for the object to remain stationary.

Normal Force and Friction

The normal force (N) is always perpendicular to the contact surface. On an incline at angle θ:

Friction force depends on N and the friction coefficient (μ). Static friction acts on objects at rest and can range from zero up to a maximum of μ_s × N. Kinetic friction acts on sliding objects with a constant magnitude of μ_k × N. Friction always opposes relative motion or the tendency to move.

When Does the Block Start to Slide?

A block will remain at rest on an incline as long as static friction can balance the gravitational component along the plane. The condition for equilibrium is:

This means sliding begins when the angle exceeds arctan(μ_s). This critical angle is called the angle of repose. For steeper angles, an external force may be needed to prevent sliding.

Moving Up or Down at Constant Speed

When an object slides at constant velocity (zero acceleration), the net force along the plane is zero. The required applied force depends on the direction of motion:

• Constant speed up: Applied force must overcome both gravity and kinetic friction (both oppose upward motion).
  F_app = mg·sin(θ) + μ_k·mg·cos(θ)
• Constant speed down: Applied force must balance the excess of gravity over friction (friction acts up-slope during downward motion).
  F_app = mg·sin(θ) - μ_k·mg·cos(θ)

How to Use This Calculator

1. Enter the incline angle (θ), mass (m), and friction coefficients (μ_s and/or μ_k) for your scenario.
2. Choose a solve mode:
   • Analyze given push — provide an applied force and see if the block moves.
   • Min force to prevent sliding — find the smallest up-slope force to keep the block at rest.
   • Min force to start sliding up — find the force threshold to begin upward motion.
   • Force for constant speed up/down — find the applied force for zero-acceleration sliding.
3. Click "Calculate" to see normal force, friction, net force, acceleration, and step-by-step reasoning.

Limitations & Assumptions

• Rigid block with no rolling or tipping.
• Friction coefficient is constant (real surfaces may vary with speed, temperature, or wear).
• Applied force is parallel to the plane surface.
• No air resistance or other external forces.
• 2D model only—no side-to-side motion.
• Educational use only—do not use for real-world ramp design, cargo loading, or safety-critical applications without professional engineering review.