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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.
A particle of mass $m$ and charge $(-q)$ enters the region between the two charged plates initially moving along the x-axis with speed $v_x$ (as shown in the figure). The length of the plate is $L$ and a uniform electric field $E$ is maintained between the plates. The vertical deflection of the particle at the far edge of the plate 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:
Liquid oxygen at $50 \text{ K}$ is heated up to $300 \text{ K}$ at a constant pressure of $1 \text{ atm}$. The rate of heating is constant. Which one of the following graphs represents the variation of temperature with time?
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:
A black body at $1227^\circ\text{C}$ emits radiations with maximum intensity at a wavelength of $5000 \text{ \AA}$. If the temperature of the body is increased by $1000^\circ\text{C}$, the maximum intensity will be observed at:
A rigid ball of mass $M$ strikes a rigid wall at $60^{\circ}$ and gets reflected without loss of speed, as shown in the figure. The value of the impulse imparted by the wall on the ball will be:
The force $F$ acting on a particle of mass $m$ is indicated by the force-time graph shown below. The change in momentum of the particle over the time interval from $0$ to $8$ s is:
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 rod PQ of mass $M$ and length $L$ is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in the figure. When the string is cut, the initial angular acceleration of the rod 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:
A solid sphere of mass $m$ and radius $R$ is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation ($E_{\text{sphere}}/E_{\text{cylinder}}$) will be:
A block of mass $m$ is in contact with the cart $C$ as shown in the figure. The coefficient of static friction between the block and the cart is $\mu$. The acceleration $a$ of the cart that will prevent the block from falling satisfies:
A particle moving with velocity $\vec{v}$ is acted by three forces shown by the vector triangle $PQR$. The velocity of the particle will:
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 wheel is subjected to uniform angular acceleration about its axis. Initially, its angular velocity is zero. In the first $2\text{ s}$, it rotates through an angle $\theta_1$. In the next $2\text{ s}$, it rotates through an additional angle $\theta_2$. The ratio of $\frac{\theta_2}{\theta_1}$ is:
Find the torque about the origin when a force of $3\hat{j} \text{ N}$ acts on a particle whose position vector is $2\hat{k} \text{ m}$.
If the acceleration due to gravity at a height 1 km above the earth is similar to a depth d below the surface of the earth, then:
A block of mass 4 kg is suspended through two light spring balances A and B connected in series. Then A and B will read respectively:
From a circular ring of mass $M$ and radius $R$, an arc corresponding to a $90^\circ$ sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is $K$ times $MR^2$. The value of $K$ will be: