When a particle of mass
Question: When a particle of mass $m$ is attached to a vertical spring of spring constant $\mathrm{k}$ and released, its motion is described by $y(t)=y_{0} \sin ^{2} \omega t$, where ' $y$ ' is measured from the lower end of unstretched spring. Then $\omega$ is :$\sqrt{\frac{\mathrm{g}}{\mathrm{y}_{0}}}$$\sqrt{\frac{g}{2 y_{0}}}$$\frac{1}{2} \sqrt{\frac{g}{y_{0}}}$$\sqrt{\frac{2 g}{y_{0}}}$Correct Option: , 2 Solution: $\mathrm{y}=\mathrm{y}_{0} \sin ^{2} \omega \mathrm{t}$ $y=\frac{y_{0}}{2}(1-...
Read More →A block of mass m attached to massless spring is
Question: A block of mass $m$ attached to massless spring is performing oscillatory motion of amplitude 'A' on a frictionless horizontal plane. If half of the mass of the block breaks off when it is passing through its equilibrium point, the amplitude of oscillation for the remaining system become $f \mathrm{~A}$. The value of $f$ is:$\frac{1}{2}$$\sqrt{2}$1$\frac{1}{\sqrt{2}}$Correct Option: , 4 Solution: At equilibrium position $V_{0}=\omega_{0} A=\sqrt{\frac{K}{m}} A$......(i) $V=\omega A^{\p...
Read More →If the potential energy between two molecules
Question: If the potential energy between two molecules is given by $\mathrm{U}=\frac{\mathrm{A}}{\mathrm{r}^{6}}+\frac{\mathrm{B}}{\mathrm{r}^{12}}$, then at equilibrium, separation between molecules, and the potential energy are : $\left(\frac{\mathrm{B}}{\mathrm{A}}\right)^{1 / 6}, 0$$\left(\frac{\mathrm{B}}{2 \mathrm{~A}}\right)^{1 / 6},-\frac{\mathrm{A}^{2}}{2 \mathrm{~B}}$$\left(\frac{2 B}{A}\right)^{1 / 6},-\frac{A^{2}}{4 B}$$\left(\frac{2 B}{A}\right)^{1 / 6},-\frac{A^{2}}{2 B}$Correct O...
Read More →A student measuring the diameter of a pencil of circular cross-section
Question: A student measuring the diameter of a pencil of circular cross-section with the help of a vernier scale records the following four readings $5.50 \mathrm{~mm}, 5.55 \mathrm{~mm}, 5.45 \mathrm{~mm} ; 5.65 \mathrm{~mm}$. The average of these four readings is $5.5375 \mathrm{~mm}$ and the standard deviation of the data is $0.07395 \mathrm{~mm}$. The average diameter of the pencil should therefore be recorded as :$(5.5375 \pm 0.0739) \mathrm{mm}$$(5.538 \pm 0.074) \mathrm{mm}$$(5.54 \pm 0....
Read More →The linear mass density of a thin rod
Question: The linear mass density of a thin rod $\mathrm{AB}$ of length $\mathrm{L}$ varies from $\mathrm{A}$ to $\mathrm{B}$ as $\lambda(\mathrm{x})=\lambda_{0}\left(1+\frac{\mathrm{x}}{\mathrm{L}}\right)$, where $\mathrm{x}$ is the distance from A. If $\mathrm{M}$ is the mass of the rod then its moment of inertia about an axis passing through $\mathrm{A}$ and perpendicular to the rod is:$\frac{5}{12} \mathrm{ML}^{2}$$\frac{3}{7} \mathrm{ML}^{2}$$\frac{2}{5} \mathrm{ML}^{2}$$\frac{7}{18} \mathr...
Read More →To raise the temperature of a certain mass of gas by
Question: To raise the temperature of a certain mass of gas by $50^{\circ} \mathrm{C}$ at a constant pressure, 160 calories of heat is required. When the same mass of gas is cooled by $100^{\circ} \mathrm{C}$ at constant volume, 240 calories of heat is released. How many degrees of freedom does each molecule of this gas have (assume gas to be ideal) ?5367Correct Option: , 3 Solution: $\mathrm{nC}_{p}(50)=160$ $\mathrm{nC}_{\mathrm{v}}(100)=240$ $\Rightarrow \frac{C_{p}}{2 C_{\gamma}}=\frac{160}{...
Read More →Two coherent sources of sound,
Question: Two coherent sources of sound, $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$, produce sound waves of the same wavelength, $\lambda=1 \mathrm{~m}$, in phase. $S_{1}$ and $S_{2}$ are placed $1.5 \mathrm{~m}$ apart (see fig.) A listener, located at $\mathrm{L}$, directly in front of $\mathrm{S}_{2}$ finds that the intensity is at a minimum when he is $2 \mathrm{~m}$ away from $\mathrm{S}_{2}$. The listener moves away from $S_{1}$, keeping his distance from $S_{2}$ fixed. The adjacent maximum of i...
Read More →Solve this following
Question: An $\mathrm{AC}$ circuit has $\mathrm{R}=100 \Omega, \mathrm{C}=2 \mu \mathrm{F}$ and $\mathrm{L}=80 \mathrm{mH}$, connected in series. The quality factor of the circuit is :$0.5$220400Correct Option: , 2 Solution: $\mathrm{Q}=\frac{1}{\mathrm{R}} \sqrt{\frac{\mathrm{L}}{\mathrm{C}}}=\frac{1}{100} \sqrt{\frac{80 \times 10^{-3}}{2 \times 10^{-6}}}$ $=\frac{1}{100} \sqrt{40 \times 10^{3}}$ $=\frac{200}{100}=2$...
Read More →If a semiconductor photodiode can detect a photon
Question: If a semiconductor photodiode can detect a photon with a maximum wavelength of $400 \mathrm{~nm}$, then its band gap energy is: $2.0 \mathrm{eV}$$1.5 \mathrm{eV}$$3.1 \mathrm{eV}$$1.1 \mathrm{eV}$Correct Option: , 3 Solution: $\Delta \mathrm{E}=\frac{\lambda \mathrm{c}}{\lambda \mathrm{e}}=3 . \mathrm{leV}$...
Read More →In the figure shown, the current in the
Question: In the figure shown, the current in the $10 \mathrm{~V}$ battery is close to: $0.36$ A from negative to positive terminal.$0.71$ A from positive to negative terminal.$0.21$ A from positive to negative terminal.$0.42$ A from positive to negative terminal.Correct Option: , 3 Solution: $\mathrm{E}_{\mathrm{eq}}=\frac{20 \times 10}{17}=\frac{200}{17}$ and $\mathrm{R}_{\text {eq }}=\frac{7 \times 10}{17}=\frac{70}{17}$ $\therefore \quad I=\frac{\frac{20}{17}-10}{4+\frac{70}{17}}=0.21 \mathr...
Read More →A particle is moving unidirectionally on a horizontal plane under
Question: A particle is moving unidirectionally on a horizontal plane under the action of a constant power supplying energy source. The displacement (s) - time (t) graph that describes the motion of the particle is (graphs are drawn schematically and are not to scale) :Correct Option: , 3 Solution: $\frac{\mathrm{dK}}{\mathrm{dE}}=\mathrm{P}=\cos \mathrm{t} \Rightarrow \mathrm{K}=\mathrm{Pt}=\frac{1}{2} \mathrm{mV}^{2}$ $\therefore \mathrm{V}=\sqrt{\frac{2 \mathrm{Pt}}{\mathrm{m}}}=\frac{\mathrm...
Read More →Shown in the figure is a hollow icecream cone
Question: Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is M, radius of its top, $\mathrm{R}$ and height, $\mathrm{H}$, then its moment of inertia about its axis is: $\frac{\mathrm{MR}^{2}}{2}$$\frac{\mathrm{MH}^{2}}{3}$$\frac{\mathrm{MR}^{2}}{3}$$\frac{M\left(R^{2}+H^{2}\right)}{4}$Correct Option: 1 Solution: Area $=\pi R \ell=\pi R\left(\sqrt{\mathrm{H}^{2}+\mathrm{R}^{2}}\right)$ Area of element $\mathrm{d} \mathrm{A}=2 \pi \mathrm{rd} \ell==2 \pi \mathrm{...
Read More →Two identical electric point dipoles have dipole moments
Question: Two identical electric point dipoles have dipole moments $\vec{p}_{1}=p \hat{i}$ and $\vec{p}_{2}=-p \hat{i}$ and are held on the $x$ axis at distance 'a' from each other. When released, they move along the $x$-axis with the direction of their dipole moments remaining unchanged. If the mass of each dipole is'm', their speed when they arc infinitely far apart is:$\frac{p}{a} \sqrt{\frac{1}{\pi \varepsilon_{0} m a}}$$\frac{\mathrm{p}}{\mathrm{a}} \sqrt{\frac{3}{2 \pi \varepsilon_{0} \mat...
Read More →The mass density of a planet of radius
Question: The mass density of a planet of radius $\mathrm{R}$ varies with the distance $r$ from its centre as $\rho(r)=\rho_{0}\left(1-\frac{r^{2}}{R^{2}}\right) .$ Then the gravitational field is maximum at:$r=\frac{1}{\sqrt{3}} R$$r=\sqrt{\frac{5}{9}} R$$r=\sqrt{\frac{3}{4}} R$$r=R$Correct Option: , 2 Solution: E $4 \pi r^{2}=\int \rho_{0} 4 \pi r^{2} d r$ $\Rightarrow \mathrm{Er}^{2}=4 \pi \mathrm{G} \int_{0}^{r} \rho_{0}\left(1-\frac{\mathrm{r}^{2}}{\mathrm{R}^{2}}\right) \mathrm{r}^{2} \mat...
Read More →Given the masses of various atomic particles
Question: Given the masses of various atomic particles $\mathrm{m}_{\mathrm{p}}=1.0072 \mathrm{u}, \mathrm{m}_{\mathrm{n}}=1.0087 \mathrm{u}$, $\mathrm{m}_{\mathrm{e}}=0.000548 \mathrm{u}, \mathrm{m}_{\overline{\mathrm{v}}}=0, \mathrm{~m}_{\mathrm{d}}=2.0141 \mathrm{u}$, where $\mathrm{p} \equiv$ proton, $\mathrm{n} \equiv$ neutron, $\mathrm{e} \equiv$ electron, $\bar{v} \equiv$ antineutrino and $d \equiv$ deuteron. Which of the following process is allowed by momentum and energy conservation?$\...
Read More →The acceleration due to gravity on the earth's surface
Question: The acceleration due to gravity on the earth's surface at the poles is $\mathrm{g}$ and angular velocity of the earth about the axis passing through the pole is $\omega$. An object is weighed at the equator and at a height $\mathrm{h}$ above the poles by using a spring balance. If the weights are found to be same, then $h$ is : ( $hR$, where $R$ is the radius of the earth)$\frac{\mathrm{R}^{2} \omega^{2}}{8 \mathrm{~g}}$$\frac{\mathrm{R}^{2} \omega^{2}}{4 \mathrm{~g}}$$\frac{R^{2} \ome...
Read More →Solve this following
Question: An electron is moving along $+\mathrm{x}$ direction with a velocity of $6 \times 10^{6} \mathrm{~ms}^{-1}$. It enters a region of uniform electric field of $300 \mathrm{~V} / \mathrm{cm}$ pointing along $+\mathrm{y}$ direction. The magnitude and direction of the magnetic field set up in this region such that the electron keeps moving along the $x$ direction will be: $5 \times 10^{-3} \mathrm{~T}$, along $+\mathrm{z}$ direction$3 \times 10^{-4} \mathrm{~T}$, along $-\mathrm{z}$ directio...
Read More →Hydrogen ion and singly ionized helium atom are accelerated,
Question: Hydrogen ion and singly ionized helium atom are accelerated, from rest, through the same potential difference. The ratio of final speeds of hydrogen and helium ions is close to:$5: 7$$1: 2$$10: 7$$2: 1$Correct Option: , 4 Solution: $\mathrm{q} \Delta \mathrm{V}=\frac{1}{2} \mathrm{mV}^{2} \Rightarrow \mathrm{v}=\sqrt{\frac{2 \mathrm{q} \Delta \mathrm{V}}{\mathrm{m}}}$ $\therefore \frac{V_{1}}{V_{2}}=\sqrt{\frac{e}{m} \frac{4 m}{e}}=2$...
Read More →Consider the force
Question: Consider the force $\mathrm{F}$ on a charge ' $\mathrm{q}$ ' due to a uniformly charged spherical shell of radius $R$ carrying charge Q distributed uniformly over it. Which one of the following statements is true for $\mathrm{F}$, if ' $\mathrm{q}$ ' is placed at distance $\mathrm{r}$ from the centre of the shell ?$\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{Qq}}{\mathrm{r}^{2}}$ for $\mathrm{r}\mathrm{R}$$\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{q} \mathrm{Q}}{\mathr...
Read More →Concentric metallic hollow spheres of radii
Question: Concentric metallic hollow spheres of radii $\mathrm{R}$ and $4 \mathrm{R}$ hold charges $\mathrm{Q}_{1}$ and $\mathrm{Q}_{2}$ respectively. Given that surface charge densities of the concentric spheres are equal, the potential difference $V(R)-V(4 R)$ is:$\frac{3 Q_{1}}{16 \pi \varepsilon_{0} R}$$\frac{Q_{2}}{4 \pi \varepsilon_{0} R}$$\frac{3 Q_{1}}{4 \pi \varepsilon_{0} R}$$\frac{3 Q_{2}}{4 \pi \varepsilon_{0} R}$Correct Option: 1, Solution: $\mathrm{E}=\frac{\mathrm{KQ}_{1}}{\mathrm...
Read More →Solve this following
Question: An electron, a doubly ionized helium ion $\left(\mathrm{He}^{++}\right)$and a proton are having the same kinetic energy. The relation between their respective de-Broglie wavelengths $\lambda_{e}, \lambda_{\mathrm{He}^{+}}$and $\lambda_{\mathrm{P}}$ is:$\lambda_{e}\lambda_{P}\lambda_{H e^{++}}$$\lambda_{\mathrm{e}}\lambda_{\mathrm{He}^{+}}=\lambda_{\mathrm{P}}$$\lambda_{\mathrm{e}}\lambda_{\mathrm{He}^{++}}\lambda_{\mathrm{P}}$$\lambda_{\mathrm{e}}\lambda_{\mathrm{P}}\lambda_{\mathrm{He...
Read More →A particle moving in the
Question: A particle moving in the $x y$ plane experiences a velocity dependent force $\overrightarrow{\mathrm{F}}=\mathrm{k}\left(\mathrm{v}_{\mathrm{y}} \hat{\mathrm{i}}+\mathrm{v}_{x} \hat{\mathrm{j}}\right)$, where $\mathrm{v}_{\mathrm{x}}$ and $\mathrm{v}_{\mathrm{y}}$ are the $\mathrm{x}$ and $\mathrm{y}$ components of its velocity $\overrightarrow{\mathrm{v}}$. If $\overrightarrow{\mathrm{a}}$ Ls the acceleration of the particle, then which of the following statements is true for the part...
Read More →The radius of R of a nucleus of mass number
Question: The radius of $R$ of a nucleus of mass number $\mathrm{A}$ can be estimated by the formula $\mathrm{R}=\left(1.3 \times 10^{-15}\right) \mathrm{A}^{1 / 3} \mathrm{~m}$. It follows that the mass density of a nucleus is of the order of: $\left(\mathrm{M}_{\text {port. }} \cong \mathrm{M}_{\text {neut. }} \simeq 1.67 \times 10^{-27} \mathrm{~kg}\right)$$10^{24} \mathrm{~kg} \mathrm{~m}^{-3}$$10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$$10^{17} \mathrm{~kg} \mathrm{~m}^{-3}$$10^{10} \mathrm{~kg} ...
Read More →A circuit to verify Ohm's law
Question: A circuit to verify Ohm's law uses ammeter and voltmeter in series or parallel connected correctly to the resistor. In the circuit:ammeter is always connected series and voltmeter in parallel.Both, ammeter and voltmeter mast be connected in series.Both ammeter and voltmeter must be connected in parallel.ammeter is always used in parallel and voltmeter is series.Correct Option: 1 Solution: Conceptual Option (1) is correct Ammeter :- In series connection, the same current flows through a...
Read More →An infinitely long straight wire carrying current I,
Question: An infinitely long straight wire carrying current I, one side opened rectangular loop and a conductor $\mathrm{C}$ with a sliding connector are located in the same plane, as shown in the figure. The connector has length $l$ and resistance R. It slides to the right with a velocity v. The resistance of the conductor and the self inductance of the loop are negligible. The induced current in the loop, as a function of separation $r$, between the connector and the straight wire is : $\frac{...
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