Ncert Exercises & Solutions

The figure shows a lamina in XY-plane. Two axes z and z' pass perpendicular to its plane. A force $\vec{F}$ acts in the plane of the lamina at point P as shown. (The point P is closer to the z'-axis than the z-axis.)

a.	torque $\vec{\tau}$ caused by $\vec{F}$ about z-axis is along - $\hat{k}$
b.	torque $\vec{\tau}'$ caused by $\vec{F}$ about z'-axis is along - $\hat{k}$
c.	torque caused by $\vec{F}$ about the z-axis is greater in magnitude than that about the z'-axis
d.	total torque is given by $\vec{\tau}_{net}=\vec{\tau}+\vec{\tau}'$

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

(3) Hint: The direction of the torque depends on the direction of the force and the magnitude f the torque depends on the distance of the force from the axis.

Step 1: Find the direction of the torque about the two axes.

(a) Consider the adjacent diagram, where r > r'.
Torque $\vec{τ}$ about z-axis = r x F which is along $\hat{k}$

(b) ${\vec{τ}}^{'} = {\vec{r}}^{'} \times \vec{F} which is along - \hat{k}$

Step 2: Find the magnitude of the torque about the two axes.

(c) $| \vec{τ} |_{z} = {Fr}_{⊥} =$ the magnitude of the torque about the z-axis where $r_{⊥}$ is the perpendicular distance between F and z-axis.
$\begin{array}{l} Similarly, | \vec{τ} |_{z^{'}} = {Fr}_{⊥}^{'} \\ Clearly, r_{⊥} > r_{⊥}^{'} \Rightarrow | \vec{τ} |_{z} > | \vec{τ} |_{z^{'}} \end{array}$

(d) We are always calculating resultant torque about a common axis.

Hence, total torque ${\vec{τ}}_{net} \neq \vec{τ} + {\vec{τ}}^{'}$ , because $\vec{τ}$ and $\vec{τ}$ ' are not about a common axis.

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a.	torque \(\vec{\tau}\) caused by \(\vec{F}\) about z-axis is along - \(\hat{k}\)
b.	torque \(\vec{\tau}'\) caused by \(\vec{F}\) about z'-axis is along - \(\hat{k}\)
c.	torque caused by \(\vec{F}\) about the z-axis is greater in magnitude than that about the z'-axis
d.	total torque is given by \(\vec{\tau}_{net}=\vec{\tau}+\vec{\tau}'\)