91Ó°ÊÓ

Torque is a fundamental concept in physics that describes the rotational effect of a force applied at a distance from a pivot point or axis. Picture it as the twist you feel when you turn a doorknob. This rotational effect can be calculated using the formula:

• \( \tau = r \times F \)
• where \( \tau \) is the torque, \( r \) is the distance from the pivot point, and \( F \) is the applied force.

If you imagine a wrench turning a bolt, the longer the wrench (distance \( r \)), or the stronger you push/pull (force \( F \)), the greater the torque.
In our problem, two torques are applied, each in different directions, which is crucial because direction affects the overall torque.
Torque is directional, meaning it has both magnitude and direction. It is a vector quantity.

Angular momentum, denoted by \( \vec{\ell} \), is a measure of the quantity of rotation a particle has. When the angular momentum of an object changes over time, it is due to the presence of torque. This relationship is described by the equation:

• \( \vec{\tau} = \frac{d \vec{\ell}}{dt} \)

This expression means that the torque \( \vec{\tau} \) exerted on a particle results in the particle's angular momentum \( \vec{\ell} \) changing at a rate given by \( \frac{d \vec{\ell}}{dt} \).
In simpler terms, the bigger the torque, the faster the change in angular momentum.
In the exercise, two torques are applied, totaling \(2.0 \hat{i} - 4.0 \hat{j}\) N·m. This combined torque tells us how the particle's angular momentum is changing every second.
Understanding this relationship aids in predicting how objects will move when forces are applied.

Vector addition is essential in calculating the total effect when multiple forces or torques act on a point. Each torque in the problem is represented as a vector because it has both direction and magnitude.
In vector notation, each component (e.g., \( \hat{i} \) and \( \hat{j} \)) represents a directional axis, like \( x \) and \( y \), and helps to understand how forces or torques overlap or counteract each other.
For instance, in the original exercise, \( \vec{\tau}_{1} = 2.0 \hat{i} \) N·m and \( \vec{\tau}_{2} = -4.0 \hat{j} \) N·m. Adding these vectors gives the total torque:

• \( \vec{\tau} = 2.0 \hat{i} + (-4.0) \hat{j} = 2.0 \hat{i} - 4.0 \hat{j} \)

This highlights how each torque's directional influence is combined to find the net effect. Properly summing these vectors is key in determining how the angular momentum of the system is changing.