Optimal Angle of Applied Force

The result

To move a body on a rough horizontal surface with the least possible force, apply the force at angle $\theta_{opt}$ above the horizontal:

$\theta_{opt} = \tan^{-1}(\mu)$

This is exactly the angle of friction $\lambda = \tan^{-1}(\mu)$.

Derivation

Derived in the Minimum Force entry. The force required at angle $\alpha$:

$F = \frac{\mu mg}{\cos\alpha + \mu\sin\alpha}$

Minimising $F$ means maximising the denominator. Setting $\frac{d}{d\alpha}(\cos\alpha + \mu\sin\alpha) = 0$:

$-\sin\alpha + \mu\cos\alpha = 0 \implies \tan\alpha = \mu$

$\theta_{opt} = \tan^{-1}(\mu)$

Physical interpretation

The optimal angle equals the angle of friction. This has an elegant geometric meaning:

The resultant contact force (normal + friction) makes angle $\lambda = \tan^{-1}(\mu)$ with the normal to the surface.

For the body to just move, the applied force must overcome this resultant contact force. The most efficient direction to apply the force is directly opposite to the resultant contact force — which means at angle $\lambda$ above the horizontal (since the resultant contact force points at angle $\lambda$ from the vertical).

The geometry says: minimum effort when you pull directly against the obstacle.

Dependence on $\mu$

Higher friction surfaces need a steeper pulling angle to minimise effort.

Practical context

This result explains why:

• It is easier to pull a heavy suitcase tilted back toward you than dragging it flat
• Movers tilt heavy furniture at an angle when sliding it
• The optimal angle increases on rougher surfaces