Ncert Exercises & Solutions

The net external torque on a system of particles about an axis is zero. Which of the following are compatible with it?

(a)	The forces may be acting radially from a point on the axis.
(b)	The forces may be acting on the axis of rotation.
(c)	The forces may be acting parallel to the axis of rotation.
(d)	The torque caused by some forces may be equal and opposite to that caused by other forces.

Choose the correct option.
1. (a, c, d)
2. (a, b, d)
3. (a, b, c, d)
4. (b, c, d)

Hint: Use torque,

\vec{Γ} = \vec{r} \times \vec{F} = Fr \sin θ \hat{n} .

$\vec{Γ} = \vec{r} \times \vec{F} = Fr \sin θ \hat{n} .$

Step: Find the torque for different cases.

We know that torque on a system of particles $\vec{Γ} = \vec{r} \times \vec{F} = Fr \sin θ \hat{n}$ $\vec{Γ} = \vec{r} \times \vec{F} = Fr \sin θ \hat{n}$ ....(i)
where $θ$ $θ$ is the angle between $\vec{r}$ $\vec{r}$ and $\vec{F}$ $\vec{F}$ . and $\vec{n}$ $\vec{n}$ is a unit vector perpendicular to both $\vec{r}$ $\vec{r}$ and $\vec{F}$ $\vec{F}$ .

(a) When forces act radially then $θ = 0 . Therefore, | \vec{Γ} | = 0$ $θ = 0 . Therefore, | \vec{Γ} | = 0$ [from Eq. (i)]
(b) When forces are acting on the axis of rotation, $r = 0, therefore, | \vec{Γ} | = 0$ $r = 0, therefore, | \vec{Γ} | = 0$ [from Eq. (i)]
(c) When forces acting parallel to the axis of rotation $then, θ = 0^{\circ}, | \vec{Γ} | = 0$ $then, θ = 0^{\circ}, | \vec{Γ} | = 0$ [from Eq. (i)]
(d) When torque by forces are equal and opposite. Therefore, ${\vec{Γ}}_{net} = {\vec{Γ}}_{1} - {\vec{Γ}}_{2} = 0$ ${\vec{Γ}}_{net} = {\vec{Γ}}_{1} - {\vec{Γ}}_{2} = 0$
Hence, option (3) is the correct answer.

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