Animated Solution for Physics - Laws of Motion: A body of mass $1\text{ kg}$ rests on a horizontal floor with which it has a coefficient of static friction $1/\sqrt{3}$. It is desired to make the body move by applying the minimum possible force $F$ newton. The value of $F$ will be ............... . (Round off to the nearest integer) (Take, $g=10{\text{ ms}}^{-2}$)

## The Art of Laziness: Finding the Minimum Force to Move a Block

Imagine you are tasked with moving a heavy block across a rough floor. Naturally, you want to accomplish this with the absolute minimum effort possible. Your first instinct might be to push or pull it perfectly horizontally. But is that really the most efficient way? Let's dive into the physics of friction and discover a beautiful mathematical secret.

When you pull a block horizontally, you have to overcome the full force of friction, which is determined by the block's entire weight pressing down on the floor. However, if you pull the block at an upward angle $\theta$, something magical happens.

Your applied force $F$ now has two components: 1. A horizontal component, $F\cos \theta$, which actively pulls the block forward. 2. A vertical component, $F\sin \theta$, which pulls the block slightly upward.

This upward pull is the game-changer. It partially lifts the block, reducing the normal force $N$ exerted by the floor. Since friction is directly proportional to the normal force ($f=\mu N$), reducing the normal force automatically reduces the friction you have to fight against!

Let's formalize this with Newton's Laws. In the vertical direction, the block is in equilibrium. The upward forces (Normal force $N$ and the vertical component of your pull $F\sin \theta$) must balance the downward force of gravity ($mg$).

For the block to just start moving, your horizontal pull must equal the maximum static friction:

Substituting our expression for $N$ into the friction equation, we get:

Expanding and rearranging to isolate $F$:

We want to find the minimum possible value for $F$. Looking at our equation, $F$ will be minimized when the denominator, $(\cos \theta +\mu \sin \theta )$, is maximized.

From trigonometry, we know that the maximum value of any expression in the form $a\cos \theta +b\sin \theta$ is exactly $\sqrt{a^2+b^2}$. In our case, $a=1$ and $b=\mu$. Therefore, the maximum value of the denominator is $\sqrt{1^2+{\mu }^2}=\sqrt{1+{\mu }^2}$.

Substituting this maximum denominator back into our force equation gives us the legendary formula for the minimum force required to move a block:

Now, let's apply this to our specific problem. We are given: - Mass $m=1\text{ kg}$ - Gravity $g=10{\text{ ms}}^{-2}$ - Coefficient of static friction $\mu =\frac{1}{\sqrt{3}}$

Plugging these values into our master formula:

As a bonus insight, at what exact angle $\theta$ does this minimum force occur? Using calculus to maximize the denominator, we find that the optimal angle satisfies the condition $\tan \theta =\mu$. This specific angle is known in physics as the Angle of Friction.

So, the next time you need to drag a heavy box, don't just pull it horizontally. Estimate the friction, calculate the angle of friction, and pull at that exact angle. Physics guarantees it will be the easiest path!