When a tree is cut, the cut is made on the side facing the direction in which the tree is required to fall. Wh

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Q. When a tree is cut, the cut is made on the side facing the direction in which the tree is required to fall. Why?

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Concept understanding — Torque

What is Torque? The Intuition

Think about opening a door. You push near the handle, not near the hinges. Why? Because the same push is far more effective when applied farther from the hinge. That "turning effectiveness" of a force is exactly what torque measures.

If you push a door with 10 N of force right at the hinge, the door barely moves. Push with the same 10 N at the handle, and it swings open easily. The force is identical — what changed is the distance from the axis of rotation. Torque combines force and distance into one quantity that tells you how much a force will make something spin.

Note

Torque is to rotation what force is to translation. Force makes things move in a straight line; torque makes things rotate.

The Precise Definition

Torque $\vec{\tau}$ about a point (or axis) is defined as the cross product of the position vector $\vec{r}$ (from the axis to the point where force acts) and the force vector $\vec{F}$ :

\vec{\tau} = \vec{r} \times \vec{F}

The magnitude is:

\tau = rF\sin\theta

where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$ .

The direction of torque is given by the right-hand rule: curl your fingers from $\vec{r}$ toward $\vec{F}$ , and your thumb points along $\vec{\tau}$ . This direction is perpendicular to both $\vec{r}$ and $\vec{F}$ .

Tip

For maximum torque, push perpendicular to the lever arm ( $\theta = 90^\circ$ , so $\sin\theta = 1$ ). Pushing directly toward or away from the axis ( $\theta = 0^\circ$ or $180^\circ$ ) produces zero torque — that's why you can't open a door by pushing straight into it.

Why Torque Equals Rate of Change of Angular Momentum

This is the rotational analogue of Newton's second law: $\vec{F} = d\vec{p}/dt$ .

For a particle of mass $m$ at position $\vec{r}$ with velocity $\vec{v}$ , its linear momentum is $\vec{p} = m\vec{v}$ . Angular momentum about the same point is:

\vec{L} = \vec{r} \times \vec{p}

Now differentiate with respect to time:

\frac{d\vec{L}}{dt} = \frac{d\vec{r}}{dt} \times \vec{p} + \vec{r} \times \frac{d\vec{p}}{dt}

Since $d\vec{r}/dt = \vec{v}$ and $\vec{p} = m\vec{v}$ , the first term is $\vec{v} \times m\vec{v} = 0$ (cross product of parallel vectors). The second term uses $d\vec{p}/dt = \vec{F}$ , giving:

\frac{d\vec{L}}{dt} = \vec{r} \times \vec{F} = \vec{\tau}

Important

Net torque equals the rate of change of angular momentum: $\vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt}$ . This holds for a system of particles as well, provided the torque and angular momentum are taken about the same point.

A Concrete Example

A wrench tightening a bolt. You apply a force of 50 N perpendicular to a wrench that is 0.3 m long. The torque magnitude is:

\tau = (0.3\ \text{m})(50\ \text{N})\sin 90^\circ = 15\ \text{N·m}

If you instead push at a 30° angle, the effective component drops:

\tau = (0.3)(50)\sin 30^\circ = 7.5\ \text{N·m}

Half the torque — you'd need twice the force for the same tightening effect.

Watch out

Torque is a vector, but in many problems you only need its magnitude and sign (clockwise or counterclockwise). Pick a sign convention and stick to it. A common mistake is forgetting that only the perpendicular component of force contributes to torque.

Torque is introduced early in the NCERT Class 11 Physics chapter on System of Particles and Rotational Motion, and 'torque formula τ = r × F' or 'torque important questions class 11 physics' are among the most searched terms for this chapter. The link between torque and rate of change of angular momentum shown here is also a recurring JEE Main derivation-based question.

Step 1. Model the standing tree as a rigid body in rotational equilibrium.

A standing tree is in equilibrium not because no torque acts on it, but because every torque is balanced. The tree's own weight $mg$ acts vertically downward through its center of gravity, which for a symmetric, upright tree lies directly above the base — so the weight's line of action passes straight through the trunk's cross-section and produces (to first approximation) zero net torque about the base. The trunk's own strength — the wood fibres all around the base — supplies whatever small restoring torque is needed to keep the tree upright against wind, its own slight asymmetries, etc.

Step 2. See what a horizontal cut (notch) does to this balance.

A lumberjack's cut removes a wedge of wood from one side of the trunk near the base. This has two combined effects: (a) it takes away the load-bearing wood fibres on that side, so the trunk can no longer supply a restoring torque from that side, and (b) it effectively shifts the location of the remaining intact wood — the 'hinge' about which the tree can still bend — to the opposite side of the trunk, away from the cut.

Step 3. Identify which way gravity's torque now acts.

Once the cut is made, the only wood left to act as a pivot/hinge is on the side opposite the cut. Taking torques about this remaining hinge wood, the tree's weight $mg$ , acting through the center of gravity of the trunk and canopy, now has a nonzero perpendicular lever arm on the side of the cut — so it produces a genuine, unbalanced torque $\tau = mg\, r_\perp$ that tends to rotate the tree toward the cut side, i.e., in the direction the cut is facing.

Step 4. Confirm this is exactly the direction the lumberjack wants.

Since the cut is deliberately made on the side facing the direction the tree is required to fall, the unbalanced torque created by removing that wood acts precisely to rotate the tree that way. As the cut is deepened, the remaining uncut wood (a thin strip, called the 'hinge' by lumberjacks) becomes less and less able to supply a resisting torque, until it can no longer support the overturning torque of the tree's own weight, and the tree rotates about that hinge and falls in the intended direction — controlled by where the cut, not the tree's own natural lean, is placed.

✓Final answer

The cut removes the wood on the side facing the fall direction, so that side can no longer supply a restoring torque; the tree's own weight then produces an unbalanced torque about the remaining uncut wood (the hinge) that rotates and topples the tree toward the cut — this is how the direction of fall is deliberately controlled.

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Q.When a tree is cut, the cut is made on the side facing the direction in which the tree is required to fall. Why?

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