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derive equation torque in cartesian coordinates from rotation of particle in plane:physical meaning of torque

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To derive the equation for torque in Cartesian coordinates, we first need to understand what torque represents physically. Torque is a measure of the rotational force applied to an object, which causes it to rotate around an axis. It depends on both the magnitude of the force applied and the distance from the axis of rotation to the line of action of the force. Let's break this down step by step.

Defining Torque

Mathematically, torque (τ) can be defined as the cross product of the position vector (r) and the force vector (F). In Cartesian coordinates, the position vector can be expressed as:

• r = xi + yj + zk

Here, x, y, and z are the coordinates of the point where the force is applied, and i, j, k are the unit vectors in the x, y, and z directions, respectively.

Force Vector in Cartesian Coordinates

The force vector can similarly be expressed as:

• F = Fx i + Fy j + Fz k

Where Fx, Fy, and Fz are the components of the force in the x, y, and z directions. Now, we can express torque as:

Torque Equation

The torque τ can be calculated using the formula:

• τ = r × F

Using the determinant form for the cross product, we can write:

• τ = |i j k|
• |x y z|
• |Fx Fy Fz|

Calculating this determinant, we get:

• τ = (yFz - zFy)i + (zFx - xFz)j + (xFy - yFx)k

Physical Interpretation

Each component of the torque vector represents the tendency of the force to cause rotation about a specific axis:

• The i-component (yFz - zFy) indicates rotation about the x-axis.
• The j-component (zFx - xFz) indicates rotation about the y-axis.
• The k-component (xFy - yFx) indicates rotation about the z-axis.

In essence, torque not only tells us how much rotational force is being applied but also the direction in which that force will cause the object to rotate.

Example for Clarity

Imagine you are using a wrench to tighten a bolt. The distance from the bolt (the axis of rotation) to where you apply the force (the end of the wrench) is crucial. If you apply a force at the end of a long wrench, you generate more torque than if you apply the same force closer to the bolt. This illustrates how both the magnitude of the force and the distance from the pivot point influence the effectiveness of the torque.

In summary, torque is a vector quantity that encapsulates the rotational effect of a force applied at a distance from an axis. The derived equation in Cartesian coordinates provides a clear mathematical representation of this physical concept, allowing us to analyze rotational dynamics effectively.