Mechanics PYQ — Page 71

All the graphs below are intended to represent the same motion. One of them does it incorrectly. Pick it up.

The relative error in the determination of the surface area of a sphere is $\alpha$. The relative error in the determination of its volume is

A man in a car at location Q on a straight highway is moving with speed $\mathrm{v}$. He decides to reach a point $P$ in a field at a distance $d$ from highway (point $M$ ) as shown in the figure.Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach $P$ is minimum? 

As shown in the figure, forces of $10^5 \mathrm{~N}$ each are applied in opposite directions, on the upper and lower faces of a cube of side $10 \mathrm{~cm}$, shifting the upper face parallel to itself by $0.5 \mathrm{~cm}$. If the side of another cube of the same material is, $20 \mathrm{~cm}$ then under similar conditions as above, the displacement will be: 

A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area $a$ floats on the surface of the liquid, covering entire cross-section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere $(\frac{dr}{r})$ , is:

In a screw gauge, $5$ complete rotations of the screw cause it to move a linear distance of $0.25\mathrm{cm}$. There are$100$ circular scale divisions. The thickness of a wire measured by this screw gauge gives a reading of $4$ main scale divisions and $30$ circular scale divisions. Assuming negligible error, the thickness of the wire is

A uniform $\operatorname{rod} A B$ is suspended from a point $X$, at a variable distance from $x$ from $A$, as shown. To make the rod horizontal, a mass $m$ is suspended from its end $A$. A set of $(m, x)$ values is recorded. The appropriate variable that give a straight line, when plotted, are: 

A body of mass $2 \mathrm{~kg}$ slides down with an acceleration of $3 \mathrm{~m} / \mathrm{s}^2$ on a rough inclined plane having a slope of $30^{\circ}$. The external force required to take the same body up the plane with the same acceleration will be: $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\right)$

A thin circular disk is in the $xy$ plane as shown in the figure. The ratio of its moment of inertia about $z$ and $z'$ axes will be: 

The velocity-time graphs of a car and a scooter are shown in the figure. (i) the difference between the distance travelled by the car and the scooter in $15 \mathrm{~s}$ and (ii) the time at which the car will catch up with the scooter are, respectively 

Two particles of the same mass $m$ are moving in circular orbits because of force, given by $F (r)=-\frac{16}{r}-{r}^{3}$. The first particle is at a distance $r=1$, and the second, at $r=4$. The best estimate for the ratio of kinetic energies of the first and the second particle is closest to

A thin uniform bar of length $\mathrm{L}$ and mass $8 \mathrm{~m}$ lies on a smooth horizontal table. Two point masses $\mathrm{m}$ and $2 \mathrm{~m}$ moving in the same horizontal plane from opposite sides of the bar with speeds $2 \mathrm{v}$ and $v$ respectively. The masses stick to the bar after collision at a distance $\frac{\mathrm{L}}{3}$ and $\frac{\mathrm{L}}{6}$ respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be: 

A body of mass $m={10}^{-2}\mathrm{kg}$ is moving in a medium and experiences a frictional force $F=-k{v}^{2} .$ Its initial speed is ${v}_{0}=10m{s}^{-1}.$ After $10s$ its kinetic energy is $\frac{1}{8}m{v}_{0}^{2},$ then value of $k$ will be:-

A man grows into a giant such that his linear dimensions increase by a factor of $9.$ Assuming that his density remains same, the stress in the leg will change by a factor of:

The machine as shown has $2$ rods of length $1m$ connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has roller that rolls along the floor in a slot. As the roller goes back and forth, a $2\mathrm{kg}$ weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a : 

Two particles $A$ and $B$ of equal mass $M$ are moving with the same speed $v$ as shown in figure. They collide completely inelastic and move as a single particle $C$. The angle $\theta$ that the path of $C$ makes with the $X$-axis is given by- 

In a physical balance working on the principle of moments, when $5 \mathrm{mg}$ weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct?

The variation of acceleration due to gravity $g$ with distance $d$ from the centre of the earth is best represented by ($R$= Earth's radius):

The moment of inertia of a uniform cylinder of length $l$ and radius $R$ about its perpendicular bisector is $I.$ What is the ratio $l/R$ such that the moment of inertia is minimum?

If the Earth has no rotational motion, the weight of a person on the equator is $W$. Determine the speed with which the earth would have to rotate about its axis so that the person at the equator will weigh $\frac{3}{4} W$. The radius of the Earth is $6400\mathrm{km}$ and $g=10 m{s}^{-2}$

A slender uniform rod of mass $M$ and length $l$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $\theta$ with the vertical is: 

Moment of inertia of an equilateral triangular lamina $ABC$, about the axis passing through its centre $O$ and perpendicular to its plane is ${I}_{0}$ as shown in the figure. A cavity $DEF$ is cut out from the lamina, where $D,E,F$ are the mid points of the sides. Moment of inertia of the remaining part of lamina about the same axis is: 

Which graph corresponds to an object moving with a constant negative acceleration and a positive velocity?

A circular hole of radius $\frac{R}{4}$ is made in a thin uniform disc having mass and radius $R$, as shown in figure. The moment of inertia of the remaining portion of the disc about an axis passing through the point $O$ and perpendicular to the plane of the disc is-