Scalar and Vector Notation

Q.

A ball is projected with a velocity v so that its range on a horizontal plane is twice the greatest height attained by the projectile. Then, its range is

1. 

2. 

3. 

4. 

Q. If a rigid body is subjected to two forces $\overrightarrow{F_1}=2\hat{i}+3\hat{j}+4\hat{k}$ acting at $(3,3,4)$ and $\overrightarrow{F_2}=-2\hat{i}-3\hat{j}-4\hat{k}$ acting at $(1,0,0)$, then which of the following is true ?

1. The body is in equilibrium.
2. The body is under the influence of a torque only.
3. The body is under the influence of a single force.
4. The body is under the influence of a force as well as a torque.

Q. The classroom door is of width $50\text{cm}$. The force of $5\text{N}$ is applied on the handle which is situated in the middle of the smaller plate as shown below. Compute the torque about the hinge (lever arm) of the door.

1. $3.5\text{Nm}$
2. $1.5\text{Nm}$
3. $2.5\text{Nm}$
4. $2.0\text{Nm}$

Q. A force $\overrightarrow{F}=(\hat{i}–\hat{j}+\hat{k})\text{N}$ acts at point $P$. Find the torque due to the force $\overrightarrow{F}$ about $O$.

1. $\hat{i}+6\hat{j}+5\hat{k}\text{Nm}$
2. $5\hat{i}-6\hat{j}+\hat{k}\text{Nm}$
3. $5\hat{i}+6\hat{j}+\hat{k}\text{Nm}$
4. $-\hat{i}-6\hat{j}-5\hat{k}\text{Nm}$
   

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. Force $\overrightarrow{F_1}=(2\hat{i}–3\hat{j})\text{N}$ and $\overrightarrow{F_2}=(-5\hat{i}+2\hat{j})\text{N}$ act through the points whose position vectors are $\hat{k}$ and $3\hat{k}$ respectively. The vector sum of the torque due to these forces is

1. $3\hat{i}+13\hat{j}\text{Nm}$
2. $-3\hat{i}-13\hat{j}\text{Nm}$
3. $-13\hat{i}-3\hat{j}\text{Nm}$
4. $-3\hat{i}+13\hat{j}\text{Nm}$

Q. In the pulley system shown in the figure, the radii of the bigger and smaller pulleys are $2\text{m}$ and $1\text{m}$ respectively. Find the resultant torque at point $\text{O}$. (Take $g=10{\text{m/s}}^2)$

1. $200\text{N-m}$
2. $100\text{N-m}$
3. $0\text{N-m}$
4. $50\text{N-m}$

Q. A force $\overrightarrow{F}=4\hat{i}-5\hat{j}+3\hat{k}$ is acting at point $\overrightarrow{r_1}=\hat{i}+2\hat{j}+3\hat{k}$. The torque acting about the point $\overrightarrow{r_2}=3\hat{i}-2\hat{j}-3\hat{k}$ is

1. Zero
2. $42\hat{i}-30\hat{j}+6\hat{k}$
3. $42\hat{i}+30\hat{j}+6\hat{k}$
4. $42\hat{i}+30\hat{j}-6\hat{k}$

Q. A particle of mass $10\text{kg}$ is projected with a velocity $5\text{m/s}$ from a point on horizontal ground, making an angle ${37}^{\circ }$ with the horizontal. Find the total torque about the point of projection acting on the particle when it is at its maximum height.

1. $120\text{Nm}$ along $\hat{k}$ direction
2. $120\text{Nm}$ along $-\hat{k}$ direction
3. $60\text{Nm}$ along $\hat{k}$ direction
4. $60\text{Nm}$ along $-\hat{k}$ direction

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. 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. In the figure shown below, find the torque due to the force of $5\text{N}$, about point $\text{O}$ and $\text{A}$.

1. $6\text{Nm}$ along $+z$ direction, $6\text{Nm}$ along $+z$ direction.
2. $6\text{Nm}$ along $+z$ direction, $6\text{Nm}$ along $-z$ direction
3. $6\text{Nm}$ along $-z$ direction, $6\text{Nm}$ along $-z$ direction
4. $6\text{Nm}$ along $-z$ direction, $6\text{Nm}$ along $+z$ direction

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. Two gear wheels of the same thickness (as shown), but one with twice the diameter of the other, are mounted on parallel light axles far enough so that the teeth of the wheels do not mesh. The larger wheel is spun with angular velocity $\Omega$ and the wheels are then moved together so that they mesh. For the meshed rotating gears, if $\frac{Kineticenergylost}{OriginalKineticenergy}=\frac{1}{3+X}$, find the value of $X$.

1. 2
2. 12
3. 4
4. 3

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. If $\overrightarrow{F}$ be a force 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