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The orthogonal projection vector of $\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}$ on $\overrightarrow{b}=4\hat{i}-4\hat{j}+7\hat{k}$ is:

Identify the thrust among the three different forces applied on the box as shown in the figure.

Co-efficient of linear expansion of brass and steel rods are ${\alpha }_1$ and ${\alpha }_2$ respectively. Lengths of brass and steel rods are $L_1$ and $L_2$ respectively. If $(L_2-L_1)$ is maintained same at all the temperatures, which one of the following relations holds good?

For the situation described in the figure, the magnetic field changes with time according to, $B=(2.00t^3-4.00t^2+0.8)\text{T}$. What is the magnitude of electric field at point ${\text{P}}_1$ when $t=0.02\text{s}$ and $r_1=0.02\text{m}$.

A body is exerting a force of 10 N on a table. The table will exert a force of on the body.

Use Moseley's law with b = 1 to find the frequency of the $K_{\alpha }$ X -ray of La(Z=57) if the frequency of the $K_{\alpha }$ X -ray of Cu(Z-29) is known to be 1.88 $\times {10}^{18}$ Hz.

Two vibrating tuning forks produce waves given by $y_1=4\sin 500\pi t$ and $y_2=2\sin 506\pi t$. Number of beats produced per minute is

The current I in the circuit shown in figure is

A water heater raises temperature of $1\text{kg}$ of water by $30{}^{\circ }\text{C}$ in $10\text{min}$. How long will it take the same heater to raise $3\text{kg}$ of oil through the same temperature. Given that specific heat of water is twice of oil.

A block of mass $1\text{kg}$ is connected by a light string passing over two smooth pulleys placed on a smooth horizontal surface as shown. Another block of $1\text{kg}$ is connected to the other end of the string then acceleration of the system and tension in the string are

Why potential at infinity is always taken zero ?

At ${40}^{\circ }\text{C}$, a brass wire of $1\text{mm}$ radius is hung from the ceiling. A small mass, $M$ is hung from the free end of the wire. When the wire is cooled down from ${40}^{\circ }\text{C}\text{to}{20}^{\circ }\text{C}$ it regains its original length of $0.2\text{m}$. The value of $M$ is close to:

(Coefficient of linear expansion and Young's modulus of brass are ${10}^{-5}{/}^{\circ }\text{C}$ and ${10}^{11}{\text{Nm}}^{-2}$ respectively; $g=10{\text{ms}}^{-2})$

A small collar of mass $m$ is given an initial velocity of magnitude $v_o$ on the horizontal circular track fabricated from a slender rod. If the coefficient of kinetic friction is ${\mu }_k$, determine the distance travelled before the collar comes to rest. (Recognize that the frictional force depends on the net normal force).

The resultant of two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ is $\overrightarrow{R}$. If Q is doubled, the new resultant is perpendicular to P. Then R equals

An equilateral triangular loop having a resistance R and length of each side l is placed in a magnetic field which is varying at dB/dt = 2ms^-1 The induced current in the loop is,

Comprehension:



A conducting rod of mass $m$ and length $l$ is released from rest on a smooth metallic rails placed in vertical plane in a uniform horizontal magnetic field $\left(B\right)$ as shown in the figure. When rod falls, it would cut magnetic field lines and motional emf will be induced. Velocity and acceleration of rod will change with time and after a long time rod will achieve a maximum velocity named as terminal velocity.





Answer the following question based on above discussion.



$\left(iii\right)$ The velocity of the rod when it has achieved its terminal value.

$E,m,J$ and $G$ denote energy, mass, angular momentum and gravitational constant respectively.

Then the dimensions of $\frac{E{J}^2}{m^5{G}^2}$ are

When a substance is kept in a magnetic field, it gets weakly repelled by the field. Which of the following represents its susceptibility?

Maximum value of $f\left(x\right)=\sin x+\cos x$ is

What is common for the given two blocks?

Magnetic force acting on the charged particle projected perpendicular to magnetic field is proportional to r^n, where r is the radius of circular path find n

The different power generating stations and distribution systems are connected by a network called :

The equation of state for a gas is given by $(P+\frac{a}{V^2})(V-b)=RT$,

where $P$, $V$ and $T$ are pressure, volume and temperature respectively and $R$ is the universal gas constant. The dimensions of the constant $a$ is

The displacement due to a wave moving in the positive $x$-direction is given by $y=\frac{1}{(1+x^2)}$ at time $t=0$ and by $y=\frac{1}{[1+(x-1{)}^2]}$ at $t=2$ seconds, where $x$ and $y$ are in metres. The velocity of the wave in $\text{m/s}$ is:

The friction coefficient between an athelete's shoes and the ground is 0.90. Suppose a superman wears these shoses and races for 50m. There is no upper limit on his capacity of running at high speeds. (a) Find the minimum time that he will have to take in completing the 50 m starting from rst. (b) Suppose he takes exactly this minimum time to complete the 50 m, what minimum time will he take to stop?

The instantaneous values of AC and AC source in a circuit are, $i=\frac{1}{\sqrt{2}}\sin \left(100\pi t\right)$ and $E=\frac{1}{\sqrt{2}}\sin (100\pi t+\pi /3)$ respectively. The average power consumed in the circuit is

If I dropped a Metal ball of weight 10kg and a feather and If I can Imagine that there is no air resistance, then,. Which do you think will reach the ground first?

Initially space between plates is empty, now $X\%$ of volume between plates is filled with $K=4$ dielectric.. If new capacitance is 3 times of earlier one, find $X$.

The ratio of kinetic energy of a planet at perihelion and aphelion during its motion around the sun in elliptical orbit of eccentricity $e$ is