Minimum Force to Move a Body

The problem

A body of mass $m$ rests on a rough horizontal surface with coefficient of static friction $\mu$. A force $F$ is applied at angle $\alpha$ above the horizontal. What is the minimum force needed to just move the body, and at what angle $\alpha$ should it be applied?

Setting up

Forces on the body:

• Weight $mg$ downward
• Normal reaction $N$ upward
• Applied force $F$ at angle $\alpha$ above horizontal
• Limiting friction $\mu N$ opposing horizontal motion

Vertical equilibrium:

$N + F\sin\alpha = mg$

$N = mg - F\sin\alpha$

Horizontal — on verge of sliding:

$F\cos\alpha = \mu N = \mu(mg - F\sin\alpha)$

$F\cos\alpha = \mu mg - \mu F\sin\alpha$

$F\cos\alpha + \mu F\sin\alpha = \mu mg$

$F(\cos\alpha + \mu\sin\alpha) = \mu mg$

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

Finding the minimum force

To minimise $F$, maximise the denominator $f(\alpha) = \cos\alpha + \mu\sin\alpha$.

Differentiate and set to zero:

$\frac{df}{d\alpha} = -\sin\alpha + \mu\cos\alpha = 0$

$\tan\alpha = \mu$

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

This is the angle of friction $\lambda$. The optimal angle equals the angle of friction.

Minimum force

Substitute $\tan\alpha = \mu$ into $F$:

At $\alpha = \tan^{-1}(\mu)$: $\sin\alpha = \frac{\mu}{\sqrt{1+\mu^2}}$, $\cos\alpha = \frac{1}{\sqrt{1+\mu^2}}$

$\cos\alpha + \mu\sin\alpha = \frac{1}{\sqrt{1+\mu^2}} + \frac{\mu^2}{\sqrt{1+\mu^2}} = \frac{1+\mu^2}{\sqrt{1+\mu^2}} = \sqrt{1+\mu^2}$

$\boxed{F_{min} = \frac{\mu mg}{\sqrt{1+\mu^2}}}$

Why lifting the force reduces the required push

When the force is purely horizontal ($\alpha = 0$): $F = \mu mg$ — the full friction must be overcome.

When the force is tilted upward: the upward component reduces the normal force $N$, which reduces friction $\mu N$. Less friction means less force needed horizontally.

But tilting too much reduces the horizontal component — inefficient again. The optimal angle balances these two effects.

Comparison

The minimum force is always less than $\mu mg$ (the force needed when pushing horizontally).