Inclined Plane with Friction
Place a block on a ramp and see whether it slides — with every force resolved.
About this simulation
Vary the incline angle, mass, friction and any applied force; the sim resolves weight into along-slope and normal components, compares the driving force to maximum static friction, and either holds or accelerates the block. Steepen the ramp with no push and the angle of repose (arctan μ) emerges naturally.
How to use it
- Raise the Incline angle slowly with Push up-slope at zero and watch the state at the top change from 'Static — friction holds' to 'Slides down', with mg·sinθ and μN compared under it.
- Change Friction and check the repose angle printed under the comparison: the block slides once the angle passes it.
- Switch Forces between All, Along slope and None to see the weight split into mg·sinθ and mg·cosθ.
- Add a Push up-slope and watch the yellow F arrow grow: the block that slid down is held once the push is big enough, and is driven up the slope when it is bigger still.
- Try the presets 'Just below repose' and 'Just above repose' to see the sudden change at the critical angle.
What to notice
- Weight is resolved into mg sin θ along the slope and mg cos θ into the slope; the parallelogram drawn with them shows they add back to mg.
- The normal force equals mg cos θ, so it falls as the ramp gets steeper and the maximum friction μN falls with it.
- With no push, the block starts to slide when tan θ exceeds μ, and this does not depend on the mass.
- A push up the slope can hold the block for a range of values, because static friction can point either up or down the slope.
The formulas
- a = g(sin θ − μ cos θ)
- Acceleration of a block sliding down a ramp of angle θ with coefficient of friction μ and no push.
- θ_r = tan⁻¹ μ
- Angle of repose: the steepest angle at which the block can rest without sliding.
- N = mg cos θ
- Normal force from the ramp on a block of mass m.
Questions
What is the angle of repose?
The angle of repose is the largest angle of an inclined plane at which a body placed on it just stays at rest. At this angle the component of weight along the slope, mg sin θ, equals the maximum static friction, μmg cos θ, which gives tan θ = μ. In the simulation, the repose angle printed on the stage is tan⁻¹ μ, and the block starts sliding once you tilt the ramp past it.
Why does the acceleration on an inclined plane not depend on mass?
Both the force pulling the block down the slope, mg sin θ, and the friction holding it back, μmg cos θ, are proportional to the mass. When you divide the net force by m to find acceleration, the mass cancels, leaving a = g(sin θ − μ cos θ). In the simulation, change the mass and both sides of the mg·sinθ vs μN comparison scale together, so whether the block slides does not change.
What is the normal force on a block on an inclined plane?
The normal force acts perpendicular to the ramp surface. With no other force pressing on the block, it balances the component of weight into the slope, so N = mg cos θ. It is always less than mg on a slope and becomes zero for a vertical wall. In the simulation, the N arrow and the dashed mg·cosθ arrow shrink as you steepen the ramp.
What minimum force is needed to stop a block sliding down a rough incline?
If mg sin θ is greater than the maximum friction μmg cos θ, a push up the slope is needed. Friction then helps the push, so the smallest push that holds the block is F = mg(sin θ − μ cos θ). A push larger than mg(sin θ + μ cos θ) will start moving the block up the slope. In the simulation, use Push up-slope and watch whether the block stays put or starts up the ramp.