The torque about the origin O due to a force 𝐅 acting on an object with position vector 𝐫 is the

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The torque about the origin O due to a force 𝐅 acting on an...

The torque about the origin $O$ due to a force $\mathbf{F}$ acting on an object with position vector $\mathbf{r}$ is the vector quantity $\mathbf{\tau}=\mathbf{r} \times \mathbf{F}$. If several forces $\mathbf{F}_{j}$ act at positions $\mathbf{r}_{j},$ then the net torque (units: $N-m$ or $\left.l b-f t\right)$ is the sum $$ \boldsymbol{\tau}=\sum \mathbf{r}_{j} \times \mathbf{F}_{j} $$ Calculate the net torque about $O$ at $P$, assuming that a $30-\mathrm{kg}$ mass is attached at $P$ [Figure $21(\mathrm{~B})]$. The force $\mathbf{F}_{g}$ due to gravity on a mass $m$ has magnitude $9.8 m \mathrm{~m} / \mathrm{s}^{2}$ in the downward direction.

The torque about the origin $O$ due to a force $\mathbf{F}$ acting on an object with position vector $\mathbf{r}$ is the vector quantity $\mathbf{\tau}=\mathbf{r} \times \mathbf{F}$. If several forces $\mathbf{F}_{j}$ act at positions $\mathbf{r}_{j},$ then the net torque (units: $N-m$ or $\left.l b-f t\right)$ is the sum $$ \boldsymbol{\tau}=\sum \mathbf{r}_{j} \times \mathbf{F}_{j} $$
Calculate the net torque about $O$ at $P$, assuming that a $30-\mathrm{kg}$ mass is attached at $P$ [Figure $21(\mathrm{~B})]$. The force $\mathbf{F}_{g}$ due to gravity on a mass $m$ has magnitude $9.8 m \mathrm{~m} / \mathrm{s}^{2}$ in the downward direction.

Key Concepts

Torque

Torque is a measure of the tendency of a force to rotate an object about an axis or pivot. It is calculated as the cross product of the position vector (which defines the point of application of the force relative to the pivot) and the force vector. The magnitude of the torque depends on both the magnitude of the force and the perpendicular distance from the pivot to the line of action of the force.

Cross Product

The cross product of two vectors is a vector that is perpendicular to the plane formed by those vectors. In the context of torque, taking the cross product of the position vector and the force vector gives a torque vector whose magnitude indicates the strength of the rotational effect and whose direction is determined by the right-hand rule.

Gravitational Force

Gravitational force is the force exerted by gravity on an object. It is calculated by multiplying the mass of the object by the acceleration due to gravity (approximately 9.8 m/s² near the Earth's surface). This force acts downward, and when applied at a certain position, it can create a torque about a chosen pivot point.

Net Torque

Net torque is the vector sum of all individual torques acting on a body. When multiple forces act at different points, the overall rotational effect about a pivot is determined by adding the torques produced by each force. This summing of torques determines whether the object will rotate, and, if so, in which direction and with what angular acceleration.

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