Scalar and Vector Notation

Q.

The resultant of two forces $PN$ and $3N$ is a force of $7N$. If the direction of $3N$ force were reversed, the resultant would be $\sqrt{19}N$. The value of $P$ is

1. $5N$

2. $6N$

3. $3N$

4. $4N$

Q. Two forces $F_1=(2\hat{i}–5\hat{j}-6\hat{k})\text{N}$ and $F_2=(-\hat{i}+2\hat{j}-\hat{k})\text{N}$ are acting on a body at the points $(1,1,0)$ and $(0,1,2)$ respectively. Find resultant torque acting on the body about point $(-1,0,1)$ in $\left(\text{N-m}\right)$.

1. $-10\hat{i}+14\hat{j}-9\hat{k}$
2. $-14\hat{i}+5\hat{j}-9\hat{k}$
3. $14\hat{i}-9\hat{j}+10\hat{k}$
4. $-14\hat{i}+10\hat{j}-9\hat{k}$

Q. $AB,CD,EG$ are perpendicular to each other. Select the correct statement(s) :

1. Torque about the axis $EG$ is $0$
2. Torque about the axis $GH$ is $0$
3. Torque about the axis $AB$ is $(OG\times F)$
4. Torque about the axis $CD$ is $(OG\times F)$

Q.

With two forces acting at a point, the maximum effect is obtained when their resultant is $4N$. If they act at right angles, then their resultant is $3N$. Then the forces are

1. $F_1=(2+\frac{1}{2}\sqrt{3})Nand{F}_2=(2-\frac{1}{2}\sqrt{3})N$

2. $F_1=(2+\sqrt{3})Nand{F}_2=(2-\sqrt{3})N$

3. $F_1=(2+\frac{1}{2}\sqrt{2})Nand{F}_2=(2-\frac{1}{2}\sqrt{2})N$

4. $F_1=(2+\sqrt{2})Nand{F}_2=(2-\sqrt{2})N$

Q. A rod of length $20\text{cm}$ is kept along $x$ axis. Two forces $5\text{N}$ and $10\text{N}$ are applied at distances $10\text{cm}$ and $15\text{cm}$ from point $A$ as shown in figure. Find the point of application of net force from point $A$.

1. $20\text{cm}$
2. $12.5\text{cm}$
3. $15\text{cm}$
4. $17.5\text{cm}$

Q. The $20\text{cm}$ diameter disc in the figure can rotate on the axle through its center. What is the net torque about $\text{O}$ (in $\text{Nm}$)?

1. $5.06$
2. $2.94$
3. $0.06$
4. $0.94$

Q. Two forces $\overrightarrow{F_1}=2\hat{i}-5\hat{j}-6\hat{k}$ and $\overrightarrow{F_2}=-\hat{i}+2\hat{j}-\hat{k}$ are acting on a body at a point $(1,1,0)$ and $(0,1,2)$ respectively. Find the torque acting on the body about the point $(-1,0,1)$.

1. $-14\hat{i}+10\hat{j}-9\hat{k}$
2. $-11\hat{i}+10\hat{j}-12\hat{k}$
3. $-3\hat{i}+3\hat{k}$
4. $-14\hat{i}-10\hat{j}-9\hat{k}$

Q. Two uniform rods of equal lengths but different masses are rigidly joined to form an $L$- shaped body, which is then pivoted about $\text{O}$ as shown in the figure. Find the net torque about point $\text{O}$. Given $M=m\sqrt{3}$

1. $0$
2. $mgL\left(\frac{\sqrt{3}}{4}\right)$
3. $mgl\left(2\sqrt{3}\right)$
4. $mgL\left(\frac{\sqrt{3}}{2}\right)$

Q.

How do you find the vector $\mathrm{v}$ with the given magnitude of $9$ and in the same direction as $u=[2,5]$ ?

Q. A force $\overrightarrow{F}=3\hat{i}+2\hat{j}-4\hat{k}$ acts at the point $(1,-1,2)$. Find its torque about the point $(2,-1,3).$

1. $6\hat{i}-7\hat{j}+\hat{k}$
2. $2\hat{i}-7\hat{j}-2\hat{k}$
3. $2\hat{i}+7\hat{j}-2\hat{k}$
4. $6\hat{i}+7\hat{j}+\hat{k}$

Q. Two forces $\overrightarrow{F_1}=2\hat{i}-5\hat{j}-6\hat{k}$ and $\overrightarrow{F_2}=-\hat{i}+2\hat{j}-\hat{k}$ are acting on a body at a point $(1,1,0)$ and $(0,1,2)$ respectively. Find the torque acting on the body about the point $(-1,0,1)$.

1. $-14\hat{i}+10\hat{j}-9\hat{k}$
2. $-11\hat{i}+10\hat{j}-12\hat{k}$
3. $-3\hat{i}+3\hat{k}$
4. $-14\hat{i}-10\hat{j}-9\hat{k}$

Q. Calculate the net torque on the system about the point $\text{O}$ as shown in figure if $F_1=11\text{N},F_2=9\text{N},F_3=10\text{N},a=10\text{cm}$ and $b=20\text{cm}$ (All the forces are along the tangents)

1. $2.2\text{Nm}$ out of the plane
2. $3.0\text{Nm}$ into the plane
3. $2.2\text{Nm}$ into the plane
4. $3.0\text{Nm}$ out of the plane

Q. A particle having mass $m$ is projected with a speed $v$ at an angle $\alpha$ with the horizontal ground. Find the torque of the weight of the particle about the point of projection when the particle reaches the ground.

1. $2m{v}^2\sin 2\alpha$
2. $\frac{m{v}^2\sin 2\alpha }{2}$
3. $m{v}^2\sin 2\alpha$
4. $0$

Q. If $\overrightarrow{F}$force is acting on a particle having the position vector$\overrightarrow{r,}$and$\overrightarrow{\tau }$ be the torque of this force about the origin, then $\overrightarrow{\tau }.\overrightarrow{F}=0$ and$\overrightarrow{r}.\overrightarrow{\tau }=0$

1. False
2. True

Q. Angular momentum of a body is defined as the product of

1. mass and angular velocity
2. centripetal force and radius
3. linear velocity and angular velocity
4. moment of inertia and angular velocity