Scalar and Vector Notation

Q. A force $\overrightarrow{F}=2\hat{i}+3\hat{j}-\hat{k}$ acts at a point $(2,-3,1)$. Then, magnitude of the torque of this force about point $(0,0,2)$ will be

1. $6\text{units}$
2. $3\sqrt{5}\text{units}$
3. $6\sqrt{5}\text{units}$
4. None of these

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

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

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

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.

Two forces, each of magnitude $F$ have a resultant of the same magnitude $F$. The angle between the two forces is

1. $45°$

2. $120°$

3. $150°$

4. $60°$

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

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 force as well as a torque.
4. The body is under the influence of a single force.

Q.

As the perpendicular distance of the application of force increases, the torque decreases.

1. True
2. False

Q. For which of the following forces, the torque about the origin is zero, if force is applied at $\overrightarrow{r}=8\hat{i}\text{m}$?

1. $16\hat{i}\text{N}$
2. $16\hat{j}\text{N}$
3. $8\hat{k}\text{N}$
4. $8\hat{i}+16\hat{j}\text{N}$

Q. Two second after projection, a projectile is travelling in a direction inclined at ${30}^{\circ }$ to the horizontal. After 1 more
Second, it is travelling horizontally. Then $(g=10m/s^2)$

1. the velocity of projection is $20\sqrt{3}m/s$
2. the angle of projection is ${30}^{\circ }$ with horizontal
3. Both (A) and (B) are correct
4. Both (A) and (B) are wrong

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

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. 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. Derive an expression for maximum height and range of an object in projectile motion.

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. 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. 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. What is the torque of a force $F=(2\hat{i}-3\hat{j}+4k)$N acting at a point $r=(3\hat{i}+2\hat{j}+3\hat{k})$m about the origin in N-m ? (Given $\tau =r\times F$ )

1. $6\hat{i}-6\hat{j}+12\hat{k}$
2. $17\hat{i}-6\hat{j}-13\hat{k}$
3. $-6\hat{i}+6\hat{j}-2\hat{k}$
4. $-17\hat{i}+6\hat{j}+13\hat{k}$

Q. A projectile thrown with an initial speed $u$ and the angle of projection ${15}^o$ to the horizontal has a range $R$. If the same projectile is thrown at an angle of ${45}^o$ to the horizontal with speed $2u$, what will be its range?

1. $7R_1$
2. $10R_1$
3. $8R_1$
4. $9R_1$

Q. When air resistance can be neglected, the maximum range of a projectile is achieved for what launch angle? (in degrees)

1. $90$
2. $45$
3. $30$
4. $60$

Q. There is some change in length when a $33000N$ tensile force is applied on a steel rod of area of cross-section ${10}^{-3}{m}^2$ . The change of temperature required to produce the same elongation of the steel rod when heated is
$(Y=3\times {10}^{11}N/m^2,\alpha =1.1\times {10}^{-5/0}C)$

1. ${20}^{\circ }C$
2. ${15}^{\circ }C$
3. ${10}^{\circ }C$
4. $0^{\circ }C$

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. 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. An electric dipole placed with its axis in the direction of a uniform electric field experiences:

1. a force but not torque
2. a torque but no force
3. a force as well as a torque
4. neither a force nor a torque

Q. Assertion :If a particle moves with a constant velocity, the angular momentum of this particle about any point remains constant. Reason: Angular momentum has the units of Plank's constant.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect