Mechanics PYQ — Page 79

A steel wire can sustain $100 \mathrm{~kg}$ weight without breaking. If the wire is cut into two equal parts, each part can sustain a weight of

Two cars of masses $m_1$ and $m_2$ are moving in circles of radii $r_1$ and $r_2$, respectively. Their speeds are such that they make complete circles in the same time $t$. The ratio of their centripetal acceleration is

An insect crawls up a hemispherical surface very slowly. The coefficient of friction between the insect and the surface is $1 / 3$. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical, the maximum possible value of $\alpha$ so that the insect does not slip is given by 

A stone of mass $m$, tied to the end of a string, is whirled around in a circle on a horizontal frictionless table. The length of the string is reduced gradually keeping the angular momentum of the stone about the centre of the circle constant. Then, the tension in the string is given by $T=A r^n$, where $A$ is a constant, $r$ is the instantaneous radius of the circle. The value of $n$ is equal to

$N$ divisions on the main scale of a vernier calliper coincide with $(N+1)$ divisions of the vernier scale. If each division of main scale is ' $a$ ' units, then the least count of the instrument is

A point particle is held on the axis of a ring of mass $m$ and radius $r$ at a distance $r$ from its centre $C$. When released, it reaches $C$ under the gravitational attraction of the ring. Its speed at $C$ will be

The mass of a spaceship is $1000 \mathrm{~kg}$. It is to be launched from the earth's surface out into free space. The value of ' $\mathrm{g}$ ' and ' $R$ ' (radius of earth) are $10 \mathrm{~m} / \mathrm{s}^2$ and $6400 \mathrm{~km}$ respectively. The required energy for this work will be ;

This question has statement $1$ and statement $2$. Of the four choices given after the statements, choose the one that best describes the two statements. If two springs $\mathrm{S_1}$ and $\mathrm{S_2}$ of force constants $k_1$ and $k_2$, respectively, are stretched by the same force, it is found that more work is done on spring $\mathrm{S_1}$ than on spring $\mathrm{S_2}$. Statement $1$ : If stretched by the same amount, work done on $\mathrm{S}_1$, will be more than that on $\mathrm{S}_2$ Statement $2: k_1 < k_2$

A structural steel rod has a radius of $10 \mathrm{~mm}$ and length of $1.0 \mathrm{~m}$. A $100 \mathrm{kN}$ force stretches it along its length. Young's modulus of structural steel is $2 \times 10^{11} \mathrm{Nm}^{-2}$. The percentage strain is about

A goods train accelerating uniformly on a straight railway track, approaches an electric pole standing on the side of track. Its engine passes the pole with velocity $u$ and the guard's room passes with velocity $v$. The middle wagon of the train passes the pole with a velocity.

A square hole of side length $\ell$ is made at a depth of $\mathrm{h}$ and a circular hole of radius $\mathrm{r}$ is made at a depth of $4 \mathrm{~h}$ from the surface of water in a water tank kept on a horizontal surface. If $\ell< < h, r< < h$ and the rate of water flow from the holes is the same, then $r$ is equal to

A thick-walled hollow sphere has outside radius $R_0$. It rolls down an incline without slipping and its speed at the bottom is $v_0$. Now the incline is waxed, so that it is practically frictionless and the sphere is observed to slide down (without any rolling). Its speed at the bottom is observed to be $5 v_0 / 4$. The radius of gyration of the hollow sphere about an axis through its centre is

Given that $K=$ energy, $V=$ velocity, $T=$ time. If they are chosen as the fundamental units, then what is dimensional formula for surface tension?

The terminal velocity of a small sphere of radius $a$ in a viscous liquid is proportional to

A car of mass $1000 \mathrm{~kg}$ is moving at a speed of 30 $\mathrm{m} / \mathrm{s}$. Brakes are applied to bring the car to rest. If the net retarding force is $5000 \mathrm{~N}$, the car comes to stop after travelling $d \mathrm{~m}$ in $t \mathrm{~s}$. Then

Two bodies $A$ and $B$ of mass $m$ and $2 m$ respectively are placed on a smooth floor. They are connected by a spring of negligible mass. $A$ third body $C$ of mass $m$ is placed on the floor. The body $C$ moves with a velocity $v_0$ along the line joining $A$ and $B$ and collides elastically with $A$. At a certain time after the collision it is found that the instantaneous velocities of $A$ and $B$ are same and the compression of the spring is $x_0$. The spring constant $k$ will be

A projectile moving vertically upwards with a velocity of $200 \mathrm{~ms}^{-1}$ breaks into two equal parts at a height of $490 \mathrm{~m}$. One part starts moving vertically upwards with a velocity of $400 \mathrm{~ms}^{-1}$. How much time it will take, after the break up with the other part to hit the ground?

Water is flowing through a horizontal tube having cross-sectional areas of its two ends being $A$ and $A^{\prime}$ such that the ratio $A / A^{\prime}$ is 5 . If the pressure difference of water between the two ends is $3 \times$ $10^5 \mathrm{~N} \mathrm{~m}^{-2}$, the velocity of water with which it enters the tube will be (neglect gravity effects)

A satellite moving with velocity $v$ in a force free space collects stationary interplanetary dust at a rate of $\frac{d M}{d t}=\alpha v$ where $M$ is the mass (of satellite + dust) at that instant. The instantaneous acceleration of the satellite is

This question has Statement 1 and Statement 2 . Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement 1: If you push on a cart being pulled by a horse so that it does not move, the cart pushes you back with an equal and opposite force. Statement 2: The cart does not move because the force described in statement 1 cancel each other.

Assuming the earth to be a sphere of uniform density, the acceleration due to gravity inside the earth at a distance of $r$ from the centre is proportional to

A solid sphere is rolling on a surface as shown in figure, with a translational velocity $v \mathrm{~m} \mathrm{~s}^{-1}$. If it is to climb the inclined surface continuing to roll without slipping, then minimum velocity for this to happen is 

A particle of mass $\mathrm{m}$ is at rest at the origin at time $\mathrm{t}=0$. It is subjected to a force $\mathrm{F}(\mathrm{t})=\mathrm{F}_0 \mathrm{e}^{-\mathrm{bt}}$ in the $x$ direction. Its speed $v(t)$ is depicted by which of the following curves?

A particle gets displaced by $\Delta \bar{r}=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \mathrm{m}$ under the action of a force $\vec{F}=(7 \hat{i}+4 \hat{j}+3 \hat{k})$. The change in its kinetic energy is