Cross Product

Q.

Refer to figure (2-E1). Find (a) the magnitude, (b) x and y components and (c) the angle with the X - axis of the resultant of $\overrightarrow{OA},\overrightarrow{BC}$ and $\overrightarrow{DE}$.

Q. At some location on earth the horizontal component of earth's magnetic field is $18\times {10}^{-6}\text{T}$. At this location, magnetic needle of length $0.12\text{m}$ and pole strength $1.8\text{Am}$ is suspended from its mid-point using a thread, it makes ${45}^{\circ }$ angle with horizontal in equilibrium.To keep this needle horizontal, the vertical force that should be applied at one of its ends is

1. $3.6\times {10}^{-5}\text{N}$
2. $1.8\times {10}^{-5}\text{N}$
3. $1.3\times {10}^{-5}\text{N}$
4. $6.5\times {10}^{-5}\text{N}$

Q. A magnetic needle lying parallel to a magnetic field requires $W$ units of work to turn it through ${60}^{\circ }$. The torque required to maintain the needle in this position will be

1. $\sqrt{3}W$
2. $W$
3. $\frac{\sqrt{3}}{2}W$
4. $2W$

Q. A particle has an angular momentum $\overrightarrow{L}$ and a linear momentum $\overrightarrow{P}$ about a reference point $O$. If $\overrightarrow{r}$ is the position vector of particle with respect to reference point, then identify correct option.

1. $\overrightarrow{L}=\overrightarrow{P}\times \overrightarrow{r}$
2. $\overrightarrow{L}\times \overrightarrow{P}=0$
3. $|\overrightarrow{L}\times \overrightarrow{P}|=LP$
4. None of these

Q. A particle has the position vector $\overrightarrow{r}=\hat{i}-2\hat{j}+\hat{k}$ and the linear momentum $\overrightarrow{p}=2\hat{i}-\hat{j}+\hat{k}$. Then, its angular momentum about the origin is

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

Q. Two short magnetic dipoles, $m_1$ and $m_2$ each having magnetic moment of $1{\text{A-m}}^2$ are placed at points $O$ and $P$ respectively. The distance between $OP$ is $1\text{m}$. The torque experienced by the magnetic dipole $m_2$ due to the presence of $m_1$ is ______ $\times {10}^{-7}\text{N-m}$.

Q. A particle of mass $2\text{kg}$ is moving with a uniform speed of $10\text{m/s}$ in $x-y$ plane along the line $x=6\text{m}$. Find the magnitude of the angular momentum of the particle about the origin.

1. $0{\text{kg-m}}^2/\text{sec}$
2. $60{\text{kg-m}}^2/\text{sec}$
3. $120{\text{kg-m}}^2/\text{sec}$
4. $20{\text{kg-m}}^2/\text{sec}$

Q. Find the angular momentum of the particle of mass $0.05\text{kg}$, position vector $\overrightarrow{r}=(8\hat{i}+4\hat{j})\text{m}$ and moving with a velocity $2\hat{i}\text{m/s}$ about the origin.

1. $-0.4\hat{i}{\text{kg-m}}^2/\text{sec}$
2. $-0.4\hat{j}{\text{kg-m}}^2/\text{sec}$
3. $-0.4\hat{k}{\text{kg-m}}^2/\text{sec}$
4. $0.4\hat{k}{\text{kg-m}}^2/\text{sec}$

Q. Given $\overrightarrow{C}=\overrightarrow{A}\times \overrightarrow{B}$ and $\overrightarrow{D}=\overrightarrow{B}\times \overrightarrow{A}$, What is the angle between $\overrightarrow{C}and\overrightarrow{D}$?

1. zero
2. 
3. 
4. 

Q. A thin uniform rod with negligible mass and length $0.2\text{m}$ is attached to the floor by a frictionless ring at point $\text{P}$. A horizontal spring with spring constant $k=4.80{\text{Nm}}^{-1}$ connected to rod, the other end of the spring is fixed with a vertical wall as shown in the figure below. The rod is in a uniform magnetic field $B=0.340\text{T}$ directed into the plane of figure. There is a current $I=6.50\text{A}$ in the rod in the direction shown. Find the energy stored in the spring when the rod is in equilibrium:

1. $4.8\times {10}^{-3}\text{J}$
2. $3.5\times {10}^{-3}\text{J}$
3. $7.8\times {10}^{-3}\text{J}$
4. $6.2\times {10}^{-3}\text{J}$

Q. The torque and magnetic potential energy of a magnetic dipole in most stable position in a uniform magnetic field of magnetic strength $5\text{T}$, having magnetic moment $\mu =2{\text{Am}}^2$ will be

1. $10\text{N-m},+20\text{J}$
2. $20\text{N-m},-10\text{J}$
3. $0\text{N-m},-10\text{J}$
4. $0\text{N-m},+10\text{J}$

Q.

Vectors $\overrightarrow{C}$ and $\overrightarrow{D}$ have magnitudes of 3 and 4 units respectively. What is the angle between the directions of $\overrightarrow{C}$ and $\overrightarrow{D}$ if the magnitude of vector product $\overrightarrow{C}\times \overrightarrow{D}$ is x and y respectively?

(x) 0 (y) 12 units

(i) 0 (ii)$\frac{\pi }{2}$ (iii)$\frac{\pi }{3}$ (iv) $\frac{\pi }{4}$

1. (x) - (iv); (y) - (iii)

2. (x) - (i); (y) - (ii)

3. (x) - (ii); (y) - (i)

4. (x) - (i); (y) - (iv)

Q.

If $\mathrm{PQ}$ and $\mathrm{PR}$ are the two sides of a triangle, then the angle between them which gives the maximum area of the triangle is

1. $\pi$

2. $\frac{\pi }{3}$

3. $\frac{\pi }{4}$

4. $\frac{\pi }{2}$

Q. A particle of mass $m$ is projected with velocity $u$ at an angle of $\theta$ with the horizontal. The initial angular momentum of the particle about the highest point of its trajectory is equal to

1. $\frac{m{u}^{3}si{n}^2\theta cos\theta }{3g}$
2. $\frac{3m{u}^{3}si{n}^2\theta cos\theta }{2g}$
3. $\frac{m{u}^{3}si{n}^2\theta cos\theta }{2g}$
4. None of these

Q. When a bar magnet is placed in a uniform magnetic field at an angle ${30}^{\circ }$ to the direction of field, it experiences a torque of $x\text{Nm}$. The same magnet experiences a torque of $\sqrt{3}\text{Nm}$ when it makes an angle of ${45}^{\circ }$ to the field. Find the value of $x\left(\text{in Nm}\right)$.

1. $\sqrt{2}$
2. $2\sqrt{2}$
3. $\sqrt{\frac{3}{2}}$
4. $\sqrt{\frac{2}{3}}$

Q.

$\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=5$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{4}$. The area of the triangle constructed on the vectors $\overrightarrow{a}-2\overrightarrow{b}$ and $3\overrightarrow{a}+2\overrightarrow{b}$ is?

1. $50$

2. $50\sqrt{2}$

3. $\frac{50}{\sqrt{2}}$

4. $100$

Q. A magnetic needle of pole strength $20\sqrt{3}\text{Am}$ is pivoted at its center. Its $N$-pole is pulled eastward by a string. The horizontal force required to produced a deflection of ${30}^{\circ }$ from magnetic meridian $[\text{Take}{B}_H={10}^{-4}\text{T}]$ is

1. $4\sqrt{3}\times {10}^{-3}\text{N}$
2. $\frac{2}{\sqrt{3}}\times {10}^{-3}\text{N}$
3. $2\times {10}^{-3}\text{N}$
4. $4\times {10}^{-3}\text{N}$

Q. Two long parallel wires situated at a distance $2a$ are carrying equal current $i$ in opposite direction as shown in figure. The value of magnetic field at a point $P$ situated at equal distances from both the wires will be:

1. $\frac{{\mu }_{0}ia}{\pi r}$
2. $\frac{{\mu }_{0}i{a}^2}{\pi r}$
3. $\frac{{\mu }_{0}i{a}^2}{\pi r^2}$
4. $\frac{{\mu }_{0}ia}{\pi r^2}$

Q. A magnetic dipole of dipole moment $M=10{\text{Am}}^2$ is suspended at angle ${45}^{\circ }$ in an external magnetic field of $\sqrt{2}\times {10}^{-5}\text{T}$. Find the work done by the magnetic field to bring the dipole moment to be in stable equilibrium ?

1. $\sqrt{2}\times {10}^{-4}\text{J}$
2. $(1-\sqrt{2})\times {10}^{-4}\text{J}$
3. $5\times {10}^{-4}\text{J}$
4. $\frac{1}{\sqrt{2}}\times {10}^{-4}\text{J}$

Q. A particle of mass $13\text{kg}$ moves with $5\text{m/s}$ in $y-z$ plane along the path $5y=12z+60$. Here $y$ and $z$ are in metre. Magnitude of angular momentum of the particle about origin (in ${\text{kg m}}^2{\text{s}}^{-1}$), when the particle is at $z=10\text{m}$, is

[Your answer must be an integer]

Q. $M$ and $M\sqrt{3}$ are the magnetic dipole moments of the two magnets, which are joined to form a cross figure. The inclination of the system with the field, if their combination, is suspended freely in a uniform external magnetic field $B$ is

1. $\theta$ = ${30}^{\circ }$
2. $\theta$ = ${45}^{\circ }$
3. $\theta$ = ${60}^{\circ }$
4. $\theta$ = ${15}^{\circ }$

Q.

Calculate the area of the parallelogram when adjacent sides are given by the vectors $\overrightarrow{A}$ = $\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{B}$ = $2\hat{i}-3\hat{j}+\hat{k}$

1. 195 sq unit

2. sq unit

3. 0

4. sq unit

Q. A particle of mass $13\text{kg}$ moves with $5\text{m/s}$ in $y-z$ plane along the path $5y=12z+60$. Here $y$ and $z$ are in metre. Magnitude of angular momentum of the particle about origin (in ${\text{kg m}}^2{\text{s}}^{-1}$), when the particle is at $z=10\text{m}$, is

Q. An electric charge $+q$ moves with velocity $10\hat{i}\text{m/s}$ in an electromagnetic field given by $\overrightarrow{E}=3\hat{i}+\hat{j}+2\hat{k}$ and $\overrightarrow{B}=\hat{i}+\hat{j}-3\hat{k}$. The $y$-component of the force experienced by $+q$ is equal to $xq$. Find the value of $x$. (All quantities are in SI units)

Q. A conducting rod length $l$ is moved at constant velocity $v_0$ on two parallel, conducting, smooth, fixed rails, which are placed in a uniform constant magnetic field $B$ perpendicular to the plane of the rails as shown in figure. A resistance $R$ is connected between the two ends of the rail. Then which of the following is/are correct?

1. The thermal power dissipated in the resistor is equal to the rate of work done by an external person pulling the rod
2. If applied external force is doubled, then a part of the external power increases the velocity of the rod
3. If resistance $R$ is doubled, then power required to maintain the constant velocity $v_0$ becomes half.
4. Lenz's law is not satisfied if the rod is accelerated by an external force.

Q. A particle is moving in a circular path of radius $a$ under the action of an attractive potential $U=-\frac{k}{2r^2}$. Its total energy is:-

1. $\frac{k}{2a^2}$
2. $zero$
3. $-\frac{k}{4a^2}$
4. $-\frac{3k}{2a^2}$

Q. Two magnets of equal mass are joined at right angles to each other, as shown in the figure. The magnet $1$ has a magnetic moment four times that of magnet $2$. This arrangement is pivoted so that it is free to rotate in the horizontal plane. In equilibrium, what will be the angle $\left(\theta \right)$ subtended by the magnet $1$ with the magnetic meridian.

1. ${\tan }^{-1}\left(1\right)$
2. ${\tan }^{-1}\left(\frac{1}{2}\right)$
3. ${\tan }^{-1}\left(\frac{1}{4}\right)$
4. ${\tan }^{-1}\left(4\right)$

Q. Find the angular momentum of the particle of mass $0.05\text{kg}$, position vector $\overrightarrow{r}=(8\hat{i}+4\hat{j})\text{m}$ and moving with a velocity $2\hat{i}\text{m/s}$ about the origin.

1. $-0.4\hat{i}{\text{kg-m}}^2/\text{sec}$
2. $-0.4\hat{k}{\text{kg-m}}^2/\text{sec}$
3. $0.4\hat{k}{\text{kg-m}}^2/\text{sec}$
4. $-0.4\hat{j}{\text{kg-m}}^2/\text{sec}$

Q.

Show that $\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors, $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$.

Q. The area of the triangle formed by the two vectors $\overrightarrow{A}=3\hat{i}+4\hat{j}$ and $\overrightarrow{B}=-3\hat{i}+7\hat{j}$ is.

1. 33 sq. unit
2. 21 sq. unit
3. 0 sq. unit
4. $\frac{33}{2}$ sq. unit