Statement about torque in a system of particles

In summary, the statement in the book explains that for a system of particles, it is always possible to find two forces with independent torque from the chosen reference point and one force that passes through the reference point with a torque of zero. This is useful because it allows for the determination of a system's total force and torque from any reference point, and the torque remains unchanged regardless of the reference point. This is because the torque is dependent on the difference between the two forces' positions, not the specific reference point chosen.
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Hello guys, I’m studying Newton’s Mechanics at the first year of engineering, and I would like to ask one question about torque. In my book I found this statement ‘’ Given a system of particles, it’s always possible to determine a pair of forces which torque is indipedent from the pole chosen, and also a force which pass through the pole ( so its torque is 0 respect to that pole ). Unfortunately there aren't examples or a scratch of demonstration of the statement. I think I got the first part about the two forces, but I'm still a bit confused about the true utility. I know the fact that the torque of internal forces which are compatible with Newton third law is zero, and that's practically a consequence of this statement (because the torque is zero indipendently from the pole)
 
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Two forces of equal magnitude that are opposite in direction and act along different lines are called a couple. Since they are equal in magnitude but opposite in direction they give a net force of zero, but a non-zero torque. This torque is independent from the reference point as, the torque is given by
$$
\vec \tau = \vec x_1 \times \vec F - \vec x_2 \times \vec F = (\vec x_1 - \vec x_2)\times \vec F
$$
and ##\vec x_1 - \vec x_2## is independent of the reference point.

You can summarize the forces acting on a system for any reference point by the total force, acting at that point, and a torque. If the total force is zero then the torque is independent of the reference point, if not it will change when you change reference. The total force is the same regardless of the reference point.
 
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Related to Statement about torque in a system of particles

1. What is torque in a system of particles?

Torque is a measure of the rotational force applied to an object. In a system of particles, it refers to the combined torque of all the individual particles acting on the system.

2. How is torque calculated in a system of particles?

To calculate torque in a system of particles, you need to determine the force acting on each particle and its distance from the axis of rotation. The torque of each particle is then calculated using the formula: torque = force x distance from axis.

3. What is the relationship between torque and angular acceleration?

Torque and angular acceleration are directly proportional. This means that an increase in torque will result in an increase in angular acceleration, and vice versa.

4. How does the distribution of mass affect torque in a system of particles?

The distribution of mass in a system of particles can greatly impact the torque. The further the mass is from the axis of rotation, the greater the torque will be. This is because the force acting on the particle has a longer lever arm, resulting in a larger torque.

5. Can torque in a system of particles be negative?

Yes, torque in a system of particles can be negative. This occurs when the direction of the force is opposite to the direction of rotation, resulting in a clockwise rotation. Positive torque, on the other hand, results in a counterclockwise rotation.

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