Torque and Its Role in Rotation
- When you push a door near its hinges, it barely moves.
- But if you push at the edge, it swings open easily.
This difference is due to torque, the rotational equivalent of force.
What is Torque?
Definition
Torque
Torque is the rotational equivalent of force. It measures the ability of a force to cause an object to rotate.
It depends on three factors:
- Magnitude of the Force ($F$)
- Distance from the Axis ($r$): This is the lever arm, the perpendicular distance from the axis to the line of action of the force.
- Angle($\theta$): The angle between the force and the lever arm.
The formula for torque ($\tau$) is:
$$
\tau = Fr\sin\theta
$$
- Units: Torque is measured in newton-meters (Nm).
- Although this is dimensionally equivalent to a joule, torque is never expressed in joules because it is not a form of energy.
How Torque Works
- Maximum Torque:
- Occurs when the force is perpendicular to the lever arm ($\theta = 90^\circ$), making $\sin\theta = 1$.
- In this case, $\tau = Fr$.
- Zero Torque:
- If the force acts along the line of the lever arm ($\theta = 0^\circ$ or $180^\circ$), then $\sin\theta = 0$, and the torque is zero.
- Consider a wrench turning a bolt.
- Applying a force of 50 N at a distance of 0.3 m from the bolt, perpendicular to the wrench, the torque is: $$
\tau = 50 \, \text{N} \times 0.3 \, \text{m} \times \sin 90^\circ = 15 \, \text{Nm}
$$
To maximize torque, apply the force as far from the axis as possible and perpendicular to the lever arm.
Equilibrium Conditions: Translational and Rotational
For an object to be in equilibrium, it must satisfy two conditions:
- Translational Equilibrium: The net force acting on the object is zero.
- Rotational Equilibrium: The net torque acting on the object is zero.
Translational Equilibrium
Definition
Translation equilibrium
An object is in translational equilibrium when the sum of all the external forces acting on the object equals zero.
In other words, the object remains at rest or moves with constant velocity.
Mathematically, this is expressed as:
$$
\sum \vec{F} = 0
$$
Rotational Equilibrium
Definition
Rotational equilibrium
Rotational equilibrium is defined as the state of movement where angular acceleration is zero: total torque is zero.
In other words, in rotational equilibrium, the object does not rotate or rotates at a constant angular velocity.
This is expressed as:
$$
\sum \tau = 0
$$
