If you want to spin an object, you can't just apply force anywhere — it matters where you apply it and in what direction. This question leads us to torque, the rotational counterpart to force.
Definition of Torque
Torque is the measure of a force's tendency to rotate an object about an axis.
Basic Formula
If a force \( \vec{F} \) is applied at distance \( r \) from the rotation axis, perpendicular to the lever arm:
\[ \tau = rF \sin\theta \]
where:
- \( r \) = distance from axis to point of application
- \( F \) = magnitude of force
- \( \theta \) = angle between \( \vec{r} \) and \( \vec{F} \)
Special Case: Perpendicular Force
If the force is perpendicular to the lever arm (\( \theta = 90° \)):
\[ \tau = rF \]
This produces maximum torque.
Special Case: Parallel Force
If the force is parallel to the lever arm (\( \theta = 0° \) or \( 180° \)):
\[ \tau = 0 \]
Parallel forces never produce rotation.
Practical Examples
Example 1: Opening a Door
You push a door at distance r = 0.8 m from the hinge with perpendicular force F = 10 N.
\[ \tau = rF = 0.8 \times 10 = 8 \text{ N·m} \]
If you push directly on the hinge with the same force:
\[ \tau = 0 \]
The door doesn't move!
Example 2: Tightening a Bolt with a Wrench
You use a wrench of length r = 0.3 m. You apply perpendicular force F = 50 N.
\[ \tau = 0.3 \times 50 = 15 \text{ N·m} \]
If instead you press on the bolt head itself (distance 0.01 m) with the same force:
\[ \tau = 0.01 \times 50 = 0.5 \text{ N·m} \]
Much weaker!
Torque as a Vector
While the simple formula is \( \tau = rF \), torque is actually vectorial.
Right-Hand Rule
The direction of \( \vec{\tau} \) is given by the right-hand rule:
- Point your fingers in the rotation direction
- Your thumb points along \( \vec{\tau} \)
Sign Convention
- Positive: counterclockwise rotation (in 2D diagrams)
- Negative: clockwise rotation
Net Torque and Newton's Rotational Equation
If several torques act on an object, the net torque is their sum:
\[ \tau_{\text{net}} = \tau_1 + \tau_2 + \tau_3 + \cdots \]
Newton's rotational equation:
\[ \tau_{\text{net}} = I\alpha \]
where:
- \( I \) = moment of inertia
- \( \alpha \) = angular acceleration
This is exactly like \( F_{\text{net}} = ma \), but for rotation!
Example 3: Rotational Dynamics
A disk with moment of inertia I = 2 kg·m² experiences three torques:
- \( \tau_1 = +12 \) N·m (counterclockwise)
- \( \tau_2 = -5 \) N·m (clockwise)
- \( \tau_3 = +3 \) N·m (counterclockwise)
What is its angular acceleration?
Solution:
Net torque: \[ \tau_{\text{net}} = 12 - 5 + 3 = 10 \text{ N·m} \]
Angular acceleration: \[ \alpha = \frac{\tau_{\text{net}}}{I} = \frac{10}{2} = 5 \text{ rad/s}^2 \]
Torque and Power
Force changes energy; torque changes power:
\[ P = \tau \omega \]
where \( \omega \) is angular velocity.
Example: A motor produces torque τ = 100 N·m at angular velocity ω = 50 rad/s. What is its power output?
\[ P = 100 \times 50 = 5000 \text{ W} = 5 \text{ kW} \]
Torque in Rolling Motion
In rolling motion, the friction force acts at the contact point and produces torque. This torque causes angular acceleration.
\[ f \cdot R = I \alpha \]
This critical relationship links translational and rotational motion.
What You Should Know
- Definition: \( \tau = rF\sin\theta \)
- Perpendicular case: \( \tau = rF \) (maximum)
- Parallel case: \( \tau = 0 \) (ineffective force)
- Newton's law: \( \tau_{\text{net}} = I\alpha \)
- Power: \( P = \tau\omega \)
- Friction in rolling: produces torque
Preview of §11.3
Now that we know how torque spins objects, what happens if a spinning object experiences no torque? Answer: its angular momentum stays constant. This is a conservation law as important as energy conservation.
📚 See also: Halliday Vol 1, Ch 11, §11.2 — Torque and rotational dynamics. 🔗 Reference: §10.2 (Moment of inertia) — foundation of rotational equation.
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