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A circular disk of moment of inertia $I_t$ is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed $\omega_i$. Another disk of moment of inertia $I_b$ is dropped coaxially onto the rotating disk. Initially the second disk has zero angular speed. Eventually both the disks rotate with a constant angular speed $\omega_f$. The energy lost by the initially rotating disc to friction 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:

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:

What would be the torque about the origin when a force $3\hat{j} \text{ N}$ acts on a particle whose position vector is $2\hat{k} \text{ m}$?

An energy of $484 \text{ J}$ is spent in increasing the speed of a flywheel from $60 \text{ rpm}$ to $360 \text{ rpm}$. The moment of inertia of the flywheel is:

A system consists of three masses $m_1$, $m_2$ and $m_3$ connected by a string passing over a pulley P. The mass $m_1$ hangs freely and $m_2$ and $m_3$ are on a rough horizontal table (the coefficient of friction = $\mu$). The pulley is frictionless and of negligible mass. The downward acceleration of mass $m_1$ is: (Assume $m_1=m_2=m_3=m$)

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:

A force $\vec{F} = \alpha\hat{i} + 3\hat{j} + 6\hat{k}$ is acting at a point $\vec{r} = 2\hat{i} - 6\hat{j} - 12\hat{k}$. The value of $\alpha$ for which angular momentum is conserved about the origin is:

The upper half of an inclined plane of inclination $\theta$ is perfectly smooth while the lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom if the coefficient of friction between the block and lower half of the plane is given by:

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:

From a disc of radius $R$ and mass $M$, a circular hole of diameter $R$, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre?

A string is wrapped along the rim of a wheel of the moment of inertia $0.10 \text{ kg-m}^2$ and radius $10 \text{ cm}$. If the string is now pulled by a force of $10 \text{ N}$, then the wheel starts to rotate about its axis from rest. The angular velocity of the wheel after $2 \text{ s}$ will be:

The angular acceleration of a body moving along the circumference of a circle 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 $\alpha_{\text{Al}} = 2.2 \times 10^{-5} \text{ K}^{-1}$)

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 person of mass $60 \text{ kg}$ is inside a lift of mass $940 \text{ kg}$ and presses the button on the control panel. The lift starts moving upwards with an acceleration of $1.0 \text{ m/s}^2$. If $g=10 \text{ m/s}^2$, the tension in the supporting cable is:

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:

Consider a thin circular ring (A), a circular disc (B), a hollow cylinder (C) and a solid cylinder (D) of the same radii $R$ and of the same masses. If $I_A$, $I_B$, $I_C$ and $I_D$ are their moments of inertia about the axis shown, then choose the correct answer from the options given below:

If the initial tension on a stretched string is doubled, then the ratio of the initial and final speeds of a transverse wave along the string is:

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: