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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 car is negotiating a curved road of radius $R$. The road is banked at an angle $\theta$. The coefficient of friction between the tyre of the car and the road is $\mu_s$. The maximum safe velocity on this road is:
A block A of mass $m_1$ rests on a horizontal table. A light string connected to it passes over a frictionless pulley at the edge of the table and from its other end, another block B of mass $m_2$ is suspended. The coefficient of kinetic friction between block A and the table is $\mu_k$. When block A is sliding on the table, the tension in the string is:
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:
Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R) Assertion (A): Semiconservative replication was experimentally proved by Mathew Meselson and Franklin Stahl (1958) Reason (R): Meselson and Stahl used radioactive isotope 15N and equilibrium density gradient centrifugation technique. In the light of the above statements, choose the correct answer from the options given below:
The products formed when Aniline reacts with $Br_2$ water, and $NaNO_2/HCl$ will be, respectively:
Deficiency of vitamin B1 causes the disease:
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 ball is thrown vertically upwards. Which of the following graph/graphs represent velocity-time graph of the ball during its flight (air resistance is neglected)