Work done by torque
Work Done by Torque
In the realm of physics, particularly in the study of rotational motion, torque plays a crucial role. Torque can be thought of as the rotational equivalent of force in linear motion. When torque is applied to an object, it causes the object to rotate around an axis. The work done by torque is an important concept that describes the energy transferred by the torque to the object, causing it to rotate.
Understanding Torque
Before we delve into the work done by torque, let's first understand what torque is. Torque ((\tau)) is a measure of the rotational force applied to an object. It is defined as the product of the force ((F)) applied and the lever arm ((r)), which is the perpendicular distance from the axis of rotation to the line of action of the force.
The formula for torque is:
[ \tau = r \times F ]
where:
- (\tau) is the torque
- (r) is the lever arm or radius
- (F) is the force applied
- (\times) denotes the cross product, indicating that torque is a vector quantity and has both magnitude and direction.
Work Done by Torque
Work done by torque refers to the energy transferred when a torque causes an object to rotate through an angle. The formula for work ((W)) done by torque is:
[ W = \tau \cdot \theta ]
where:
- (W) is the work done by torque
- (\tau) is the torque
- (\theta) is the angle of rotation in radians
It's important to note that the angle must be in radians for the formula to be valid.
Important Points About Work Done by Torque
- Work is done only when the component of torque is in the direction of the rotation.
- If the torque is perpendicular to the direction of rotation, no work is done.
- The units of work are Joules (J) in the International System of Units (SI).
Table: Differences and Important Points
| Aspect | Linear Motion | Rotational Motion |
|---|---|---|
| Quantity | Force (F) | Torque ((\tau)) |
| Formula | (F = m \cdot a) | (\tau = r \times F) |
| Work Done | (W = F \cdot d) | (W = \tau \cdot \theta) |
| Units of Work | Joules (J) | Joules (J) |
| Directionality | Along the line of motion | Around the axis of rotation |
| Angle Measurement | Not applicable | Radians |
Examples
Example 1: Calculating Work Done by Torque
Suppose a force of 10 N is applied at the end of a wrench 0.5 m long to tighten a bolt. If the wrench is turned through a quarter of a revolution, calculate the work done by the torque.
Solution:
First, calculate the torque:
[ \tau = r \times F = 0.5 \, \text{m} \times 10 \, \text{N} = 5 \, \text{Nm} ]
Now, convert the quarter revolution to radians:
[ \theta = \frac{1}{4} \times 2\pi = \frac{\pi}{2} \, \text{radians} ]
Finally, calculate the work done:
[ W = \tau \cdot \theta = 5 \, \text{Nm} \times \frac{\pi}{2} = \frac{5\pi}{2} \, \text{J} \approx 7.85 \, \text{J} ]
Example 2: No Work Done by Torque
Imagine a scenario where a force is applied to a door knob, but the door does not move because it's locked. Even though a torque is applied, no work is done because there is no rotation ((\theta = 0)).
Solution:
Since the angle of rotation (\theta) is zero, the work done by torque is:
[ W = \tau \cdot \theta = \tau \cdot 0 = 0 \, \text{J} ]
No matter how much force is applied, if there is no rotation, the work done by torque will always be zero.
In conclusion, the work done by torque is a fundamental concept in rotational motion that describes the energy transfer when an object is rotated by a torque. It is analogous to the work done by force in linear motion and is calculated by multiplying the torque by the angle of rotation in radians. Understanding this concept is essential for analyzing and solving problems in rotational dynamics.