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A hollow sphere of diameter $0.2 \text{ m}$ and mass $2 \text{ kg}$ is rolling on an inclined plane with velocity $v = 0.5 \text{ m/s}$. The kinetic energy of the sphere is:

Match List-I (Spectral Series) with List-II (corresponding wave number expressions). List-I (Series) A. Balmer series B. Lyman series C. Brackett series D. Pfund series List-II (Wave number in cm⁻¹) I. R(1/1² - 1/n²) II. R(1/4² - 1/n²) III. R(1/5² - 1/n²) IV. R(1/2² - 1/n²) Choose the correct answer from the options given below:

A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. Four objects each of mass $m$, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be:

A constant torque of $100\text{ N m}$ turns a wheel of moment of inertia $300\text{ kg m}^2$ about an axis passing through its centre. Starting from rest, its angular velocity after $3\text{ s}$ is:

The distance covered by a body of mass $5 \text{ g}$ having linear momentum $0.3 \text{ kg m/s}$ in $5 \text{ s}$ is:

In the Bohr's model of a hydrogen atom, the centripetal force is furnished by the Coulomb attraction between the proton and the electron. If a₀ is the radius of the ground state orbit, m is the mass and e is the charge on the electron, ε₀ is the vacuum permittivity, the speed of the electron is:

Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities $\omega_1$ and $\omega_2$. They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is:

In a regular octahedral molecule, $MX_{6}$, the number of $X–M–X$ bonds at 180º is:

ABC is an equilateral triangle with O as its centre. $\vec{F}_1$, $\vec{F}_2$ and $\vec{F}_3$ represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero then the magnitude of $\vec{F}_3$ is:

A copper rod of $88 \text{ cm}$ and an aluminium rod of an unknown length have an equal increase in their lengths independent of an increase in temperature. The length of the aluminium rod is: ($\alpha_{\text{Cu}} = 1.7 \times 10^{-5} \text{ K}^{-1}$ and $\alpha_{\text{Al}} = 2.2 \times 10^{-5} \text{ K}^{-1}$)

Three identical spheres, each of mass $M$, are placed at the corners of a right-angle triangle with mutually perpendicular sides equal to $2 \text{ m}$ (see figure). Taking the point of intersection of the two mutually perpendicular sides as the origin, find the position vector of the centre of mass.

The moment of inertia of a thin uniform rod of mass $M$ and length $L$ about an axis passing through its mid-point and perpendicular to its length is $I_0$. Its moment of inertia about an axis passing through one of its ends and perpendicular to its length is:

A black body at $227^{\circ}\text{C}$ radiates heat at the rate of $7\text{ cal cm}^{-2}\text{s}^{-1}$. At a temperature of $727^{\circ}\text{C}$, the rate of heat radiated in the same units will be

A particle of mass $m$ is projected with velocity $v$ making an angle of $45^\circ$ with the horizontal. When the particle lands on the level ground, the magnitude of the change in its momentum will be:

A shell of mass $m$ is at rest initially. It explodes into three fragments having masses in the ratio $2:2:1$. If the fragments having equal masses fly off along mutually perpendicular directions with speed $v$, the speed of the third (lighter) fragment is:

The two ends of a rod of length $L$ and a uniform cross-sectional area $A$ are kept at two temperatures $T_1$ and $T_2$ ($T_1 > T_2$). The rate of heat transfer $\frac{dQ}{dt}$ through the rod in a steady state is given by:

It is easier to draw up a wooden block along a smooth inclined plane than to haul it vertically, principally because:

Two particles $A$ and $B$ initially at rest, move toward each other under the mutual force of attraction. At an instance when the speed of $A$ is $v$ and speed of $B$ is $3v$, the speed of the centre-of-mass will be:

The volume ($V$) of a monatomic gas varies with its temperature ($T$), as shown in the graph. The ratio of work done by the gas to the heat absorbed by it when it undergoes a change from state $A$ to state $B$ will be:

In an electromagnetic wave in free space the root mean square value of the electric field is $E_{rms} = 6$ V/m. The peak value of the magnetic field is: