Interactive Animation: Forces on a Wedge – Educational-Animation.org

The following animation illustrates the principle of force resolution at a wedge. The decomposition of a main force into two flank forces is visualized in real time through an interactive force parallelogram.

Description of the Animation

The animation shows a symmetrical wedge with three synchronized elements: wedge geometry, force vectors, and lines of action. The sliders allow the force and wedge angle to be changed, with the force parallelogram adjusting automatically.

The flank forces are calculated from the force equilibrium on the wedge:

\[ F_W = \frac{F}{2 \cdot \sin\left(\frac{\beta}{2}\right)} \]

With:

\( F \) – Main force (acting perpendicular to the wedge tip)
\( F_W \) – Flank force (perpendicular to the respective wedge surface)
\( \beta \) – Wedge angle (included angle at the tip)

The dashed lines of action indicate the directions of the forces and form the force parallelogram, whose geometric construction illustrates the resolution of the main force.

Interactive Controls

The following parameters can be adjusted using the sliders:

F (0–100 N): Magnitude of the main force
β (5–90°): Opening angle of the wedge

Physical Background

When a wedge penetrates a workpiece, the applied main force is resolved into two flank forces acting perpendicular to the wedge surfaces. This resolution follows the principle of the force parallelogram: the two flank forces form the sides of a parallelogram whose diagonal corresponds to the main force.

The smaller the wedge angle β, the larger the flank forces become relative to the main force. At an angle of 30°, the flank forces are already twice as large as the main force; at 10°, approximately six times as large. This amplification effect explains the enormous cutting and splitting action of sharp wedges.

Practical Applications

Cutting tools: Knives, scissors, and axes use small wedge angles for high cutting forces
Fastening technology: Nails and wedges for joining and fixing components
Machining technology: Lathe tools, milling cutters, and drills with defined cutting wedges
Splitting tools: Log splitters and stone wedges for controlled material separation

Derivation of the formula (step by step)

The wedge formula can be derived from force equilibrium on a symmetric wedge. Here is the derivation in individual steps:

Free body diagram

Consider a symmetric wedge with wedge angle \( \beta \). A force \( F \) acts from above on the back of the wedge. On each of the two wedge surfaces, a normal force \( F_W \) acts perpendicular to the surface.

Geometry

The axis of symmetry of the wedge lies along the \( y \)-direction (the direction of the applied force \( F \)). Each wedge surface is inclined by half the wedge angle \( \beta/2 \) relative to the \( y \)-axis. The normal force \( F_W \) is perpendicular to the wedge surface, so it also makes an angle of \( \beta/2 \) with the \( y \)-axis.

Force equilibrium in the y-direction

In equilibrium, the applied force \( F \) must be balanced by the \( y \)-components of the two normal forces:

\[ F = 2 \cdot F_W \cdot \sin\!\left(\frac{\beta}{2}\right) \]

The factor of 2 arises from symmetry — both wedge surfaces contribute equally.

Solving for the normal force

Solving for \( F_W \):

\[ F_W = \frac{F}{2 \cdot \sin\!\left(\frac{\beta}{2}\right)} \]

The smaller the wedge angle \( \beta \), the smaller \( \sin(\beta/2) \) becomes, and the larger the normal force \( F_W \) relative to the applied force \( F \). This is why sharp wedges are so effective.

Everyday examples with concrete numbers

The wedge formula \( F_W = \frac{F}{2 \sin(\beta/2)} \) can be illustrated with familiar tools.

Kitchen knife (\( \beta \approx 15° \))

When cutting, you press down on the knife with a force of approximately \( F = 20\;\text{N} \). The resulting normal force is:

\[ F_W = \frac{20\;\text{N}}{2 \cdot \sin(7.5°)} = \frac{20\;\text{N}}{2 \cdot 0.1305} = \frac{20\;\text{N}}{0.2611} \approx 76.6\;\text{N} \]

This represents an amplification by a factor of 3.8. A moderate pressing force on the back of the knife generates a significantly larger spreading force in the material being cut.

Splitting axe vs. splitting wedge

A splitting axe typically has a wedge angle of about \( \beta \approx 25° \), while a splitting wedge has only about \( \beta \approx 8° \). For a striking force of \( F = 50\;\text{N} \):

Splitting axe (\( \beta = 25° \)):

\[ F_W = \frac{50\;\text{N}}{2 \cdot \sin(12.5°)} = \frac{50\;\text{N}}{2 \cdot 0.2164} = \frac{50\;\text{N}}{0.4329} \approx 115.5\;\text{N} \]

Splitting wedge (\( \beta = 8° \)):

\[ F_W = \frac{50\;\text{N}}{2 \cdot \sin(4°)} = \frac{50\;\text{N}}{2 \cdot 0.0698} = \frac{50\;\text{N}}{0.1395} \approx 358.4\;\text{N} \]

With its shallower angle, the splitting wedge generates roughly three times the normal force of the splitting axe for the same striking force. However, the axe penetrates the wood more easily because its wedge surfaces offer less resistance.

Razor blade (\( \beta \approx 5° \))

A razor blade has an extremely small wedge angle. For \( F = 10\;\text{N} \):

\[ F_W = \frac{10\;\text{N}}{2 \cdot \sin(2.5°)} = \frac{10\;\text{N}}{2 \cdot 0.0436} = \frac{10\;\text{N}}{0.0872} \approx 114.6\;\text{N} \]

An amplification by a factor of 11.5 — this is why razor blades cut so effortlessly. However, the thin edge is also more fragile and wears out more quickly.

Limiting cases and extreme values

To develop a deeper understanding of the wedge effect, it is worth examining the limiting cases of the formula \( F_W = \frac{F}{2 \sin(\beta/2)} \). These can also be explored in the animation — simply move the slider to the extremes.

\( \beta \to 0° \): Infinitely thin wedge

For \( \beta \to 0° \), \( \sin(\beta/2) \to 0 \), and therefore \( F_W \to \infty \). In theory, an infinitely thin wedge could produce an arbitrarily large normal force. In practice, the material strength and elasticity of the wedge limit the achievable amplification.

\( \beta = 60° \): No amplification

\[ F_W = \frac{F}{2 \cdot \sin(30°)} = \frac{F}{2 \cdot 0.5} = F \]

At a wedge angle of 60°, the normal force equals the applied force. The wedge no longer amplifies the force; it merely redirects it.

\( \beta = 90° \): Right-angle wedge

\[ F_W = \frac{F}{2 \cdot \sin(45°)} = \frac{F}{2 \cdot 0.7071} = \frac{F}{1.4142} \approx 0.707 \cdot F \]

The normal forces are smaller than the applied force. The wedge “dilutes” the force rather than amplifying it.

\( \beta = 180° \): Flat plate

\[ F_W = \frac{F}{2 \cdot \sin(90°)} = \frac{F}{2 \cdot 1} = \frac{F}{2} \]

At \( \beta = 180° \), the wedge is a flat plate. The force is simply divided between the two sides — there is no wedge effect at all.

Symmetric vs. asymmetric wedge

The animation and the derivation above treat the symmetric wedge, where both wedge surfaces make the same angle with the axis of symmetry. In practice, however, asymmetric wedges also exist.

What changes with asymmetry?

In an asymmetric wedge — such as a single-bevel knife or a chisel — the angles of the two wedge surfaces differ. As a result, the normal forces on the two sides are no longer equal.

The steeper side (larger angle relative to the axis) produces a smaller normal force.
The shallower side (smaller angle) produces a larger normal force.

Practical significance

A chisel has one flat side (the back) and one beveled side. This causes the material to be displaced preferentially toward the bevel side. The craftsperson can thus control precisely which side the material is removed from. A symmetric wedge, by contrast, would displace material equally to both sides.

Effect of friction

The animation shows the frictionless ideal case. In reality, friction always acts on the wedge surfaces and significantly affects the wedge’s behavior.

Formula with friction

Taking friction into account with the friction coefficient \( \mu \), the friction angle \( \rho \) is defined as:

\[ \rho = \arctan(\mu) \]

\( \rho \): Friction angle
\( \mu \): Friction coefficient between wedge surface and material

The modified wedge formula then becomes:

\[ F_W = \frac{F}{2 \cdot \sin\!\left(\dfrac{\beta}{2} + \rho\right)} \]

Friction increases the effective angle in the denominator, which reduces the normal force compared to the frictionless case. Part of the applied force is lost to friction.

Self-locking

A particularly interesting effect occurs when the friction angle \( \rho \) exceeds half the wedge angle \( \beta/2 \):

\[ \rho > \frac{\beta}{2} \quad \Longleftrightarrow \quad \mu > \tan\!\left(\frac{\beta}{2}\right) \]

In this case, friction alone is sufficient to hold the wedge in place, even when no external force is applied. This phenomenon is called self-locking.

Everyday example: A door wedge with a small wedge angle and a rough surface stays in place under the door without anyone pushing against it. The friction between the wedge and the floor or door prevents it from slipping out — a simple but impressive example of self-locking.