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Torque as Cross Product

In case of mechanics, generally we discuss about the translatory motion. In case of rotational motion, only the force cannot play an important role. In translational motion, force play a major role while in case of the rotational motion, torque can play the major role. Torque is due to the couple of forces. Let us discuss about the torque and its representation in the form of cross product.



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Torque as Cross Product:

When we switch on a fan, the centre of mass of the fan remains unmoved, while the fan rotates with an angular acceleration. As the centre of mass of the fan remains at rest, the vector sum of the external forces acting on the fan must be zero. It means one can produce angular acceleration even when resultant force is zero. When an external force acting on a body has a tendency to rotate the body about a fixed point or about a fixed axis, it is said to exert a force or the torque on the body. The moment of force or the torque due to a force gives us the turning effect of the force about the fixed point or the fixed axis. It is measured by the product of the magnitude of force and perpendicular distance of the line of action of force from the axis of rotation. Torque is represented by a Greek letter. Thus, the moment of force or torque = force `xx` perpendicular distance

     `vectau` `vecr` `xx` `vecF` = r F Sin`theta` `hatn` 

Where`theta` the small angle between be`vecr` and`vecF` , `hatn`is the unit vector along`vectau`. The direction of `vectau` is perpendicular to the plane containing `vecr`and`vecF` it is determined by right hand screw rule. The SI unit of the torque is Newton metre and the dimension is [M1L2T-2].

Understanding angular momentum and torque is always challenging for me but thanks to all Physics help websites to help me out.

Conclusion for the Torque in the Form of a Cross Product:

From the above discussion, if the angle between is `theta` = 0°. Therefore, the torque t = r F Sin`theta` = r F Sin 0° = 0, turning effect of the force is zero. In case `theta` = 90°, torque t = r F Sin `theta` = r F Sin 90° = r F = the maximum value of the torque, turning effect of the force in this case is maximum. At any arbitrary angleq, the toque t = r F Sinq. By sign conventions, anticlockwise moments are taken as positive and clockwise moments are taken as negative.

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