The Inclined Plane

Gravity on an Incline

splitting the weight

On a ramp, straight-down weight splits into a part along the slope and a part into it, using the slope's own 3-4-5 triangle.

Example

On a ramp, straight-down weight is split into a part along the slope and a part into it, using the slope's own 3-4-5 triangle. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A block resting on a ramp

A block of mass 1 kilogram sits on a frictionless ramp that rises 3 for every 4 across. Gravity pulls it straight down with weight 1 times 10, which is 10 newtons.

W=m g=1 kg ⋅ 10 m/s2=10 NW = m\,g = 1\ \text{kg} \,\cdot\, 10\ \text{m}/\text{s}^{2} = \hl{10}\ \text{N}
The block and its weightA block on a ramp with a single arrow pulling straight down.mW

The slope gives the fractions

Straight-down weight is awkward on a ramp, so we split it into a part along the slope and a part pressing into it. The 3 by 4 ramp is a 5 long right triangle, so the along-slope fraction is 3 over 5 and the into-slope fraction is 4 over 5.

sin⁡θ=35,cos⁡θ=45\sin\theta = \frac{3}{5}, \quad \cos\theta = \frac{4}{5}

The part that pulls it down the slope

Multiply the weight by the along-slope fraction: 10 times 3 over 5 is 6 newtons, pointing down the slope.

W∥=Wsin⁡θ=10 N⋅35=6 NW_\parallel = W\sin\theta = 10\ \text{N} \cdot \frac{3}{5} = \hl{6}\ \text{N}

The part that presses into the slope

Multiply the weight by the into-slope fraction: 10 times 4 over 5 is 8 newtons, pressing straight into the surface.

W⊥=Wcos⁡θ=10 N⋅45=8 NW_\perp = W\cos\theta = 10\ \text{N} \cdot \frac{4}{5} = \hl{8}\ \text{N}
The weight split along and into the slopeThe downward weight arrow with its along-slope part and its into-slope part drawn from the block.mWdown-slopeinto-slope

Same ramp, larger weight, larger components

Keep the ramp fractions fixed. A larger straight-down weight makes both the down-slope pull and the into-slope press larger in the same ratio.

WW∥W⊥10 N6 N8 N20 N12 N16 N30 N18 N24 N\begin{array}{c|c|c}W & W_\parallel & W_\perp \\ \hline 10\ \text{N} & 6\ \text{N} & 8\ \text{N} \\ 20\ \text{N} & 12\ \text{N} & 16\ \text{N} \\ 30\ \text{N} & 18\ \text{N} & 24\ \text{N}\end{array}

Steeper ramp, more pull down the slope

Hold the weight fixed and change the rise fraction. A steeper ramp turns more of the same weight into down-slope pull.

risehypWW∥010 N0 N3510 N6 N4510 N8 N\begin{array}{c|c|c}\frac{\text{rise}}{\text{hyp}} & W & W_\parallel \\ \hline 0 & 10\ \text{N} & 0\ \text{N} \\ \tfrac{3}{5} & 10\ \text{N} & 6\ \text{N} \\ \tfrac{4}{5} & 10\ \text{N} & 8\ \text{N}\end{array}

Steeper ramp, less press into the slope

The same trade shows up sideways: as the rise fraction grows, the run fraction shrinks, so the into-slope press gets smaller.

runhypWW⊥110 N10 N4510 N8 N3510 N6 N\begin{array}{c|c|c}\frac{\text{run}}{\text{hyp}} & W & W_\perp \\ \hline 1 & 10\ \text{N} & 10\ \text{N} \\ \tfrac{4}{5} & 10\ \text{N} & 8\ \text{N} \\ \tfrac{3}{5} & 10\ \text{N} & 6\ \text{N}\end{array}
mechanics A 3-4-5 ramp makes the along-slope and into-slope parts exact whole numbers (a 6-8-10 split of a 10 N weight).