Figure shows a lamina in x y - plane. Two axes z and z^' pass perpendicular to its plane. A forc

step 1 So according to right -handed screw rule, right -handed screw rule, the direction of tau caused

Figure shows a lamina in x y - plane. Two axes z and z^'...

Figure shows a lamina in $x y$ - plane. Two axes $z$ and $z^{\prime}$ pass perpendicular to its plane. A force $\mathbf{F}$ acts in the plane of lamina at point $P$ as shown. Which of the following are true? (The point $P$ is closer to $z^{\prime}$-axis than the $z$-axis.) [NCERT Exemplar] (a) Torque t caused by $\mathbf{F}$ about $z$-axis is along $-\mathrm{k}$ (b) Torque $\mathrm{t}^{\prime}$ caused by $\mathrm{F}$ abut $z^{\prime}$ axis is along $-\hat{\mathbf{k}}$ (c) Torque t caused by $F$ about $z$ axis is greater in magnitude than that about $z$ axis (d) Total torque is given be $t=t+t^{\prime}$

Figure shows a lamina in $x y$ - plane. Two axes $z$ and $z^{\prime}$ pass perpendicular to its plane.
A force $\mathbf{F}$ acts in the plane of lamina at point $P$ as shown. Which of the following are true? (The point $P$ is closer to $z^{\prime}$-axis than the $z$-axis.) [NCERT Exemplar]
(a) Torque t caused by $\mathbf{F}$ about $z$-axis is along $-\mathrm{k}$
(b) Torque $\mathrm{t}^{\prime}$ caused by $\mathrm{F}$ abut $z^{\prime}$ axis is along $-\hat{\mathbf{k}}$
(c) Torque t caused by $F$ about $z$ axis is greater in magnitude than that about $z$ axis
(d) Total torque is given be $t=t+t^{\prime}$

Key Concepts

Torque (Moment) Calculation

Torque, sometimes called moment, is a measure of the tendency of a force to rotate an object about an axis. It is calculated as the cross product of the position vector, which extends from the axis of rotation to the point of application of the force, and the force vector. This definition gives both the magnitude of the torque (as the product of the force and the perpendicular distance from the axis) and its direction, which is determined by the right?hand rule.

Moment Arm (Perpendicular Distance)

The moment arm is the shortest (perpendicular) distance between the line of action of the force and the axis of rotation. Understanding the concept of the moment arm is crucial because only the component of the force that acts perpendicular to this arm contributes to the torque. A force applied further from the axis (with a longer moment arm) produces a greater torque than an equal force applied closer to the axis.

Right-Hand Rule

The right-hand rule is a convention used to determine the direction of vectors resulting from cross products, such as torque. By pointing the fingers of the right hand along the position vector and curling them toward the force vector, the thumb points in the direction of the resulting torque vector. This rule helps in establishing the correct sign and orientation of the torque with respect to a chosen coordinate system.

Force Application in a Plane and Axes Orientation

When a force is applied in a plane, its torque about an axis that is perpendicular to that plane depends on the relative geometry, specifically the distance from the force's line of action to the axis. In systems with multiple axes perpendicular to the plane, different distances (or moment arms) between the force's point of application and each axis can lead to different magnitudes and even directions (depending on conventions) of torque. This concept is essential in analyzing planar objects where forces and rotations are involved.

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