Mechanics PYQ — Page 73

In the figure shown $ABC$ is a uniform wire. If the center of mass of the wire lies vertically below point $A$, then $\frac{BC}{AB}$ is close to 

A point particle of mass m, moves along the uniformly rough track PQR as shown in the figure. The coefficient of friction, between the particle and the rough track equals $\mu$ . The particle is released, from rest, from the point P and it comes to rest at a point R. The energies, lost by the ball, over the parts, PQ and QR, of the track, are equal to each other, and no energy is lost when particle changes direction from PQ to QR. The values of the coefficient of friction $\mu$ and the distance $x=(QR)$ , are respectively close to: 

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up? Fat supplies $3.8\times {10}^{7}$ J of energy per kg which is converted to mechanical energy with a 20% efficiency rate. Take $g=9.8 m{s}^{-2}$ :

A particle of mass m is moving along the side of a square of side 'a', with a uniform speed $\upsilon$ in the x-y plane as shown in the figure:  Which of the following statements is false for the angular momentum $\vec{L}$ about the origin?

A car of weight W is on an inclined road that rises by 100 m over a distance of 1km and applies a constant frictional force $\frac{W}{20}$ on the car. While moving uphill on the road at a speed of $10 m{s}^{-1}$ , the car needs power P. If it needs power $\frac{P}{2}$ while moving downhill at speed $\upsilon$ then value of $\upsilon$ is:

A rocket is fired vertically from the earth with an acceleration of 2g, where g is the gravitational acceleration. On an inclined plane inside the rocket, making an angle $\theta$ with the horizontal, a point object of mass m is kept. The minimum coefficient of friction ${\mu }_{min}$ between the mass and the inclined surface such that the mass does not move is:

A student measures the time period of 100 oscillations of a simple pendulum four times. The data set is 90 s, 91 s, 95 s and 92 s. If the minimum division in the measuring clock is 1 s, then the reported mean time should be:

A screw gauge with a pitch of $0.5 \mathrm{mm}$ and a circular scale with $50$ divisions is used to measure the thickness of a thin sheet of aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the ${45}^{th}$ division coincides with the main scale line and that the zero of the main scale is barely visible. What is the thickness of the sheet if the main scale reading is $0.5\mathrm{mm}$ and the ${25}^{th}$ division coincides with the main scale line?

 Consider a water jar of radius R that has water filled up to height H and is kept on a stand of height h (see figure). Through a hole of radius r $(r<<R)$ at its bottom, the water leaks out and the stream of water coming down towards the ground has a shape like a funnel as shown in the figure. If the radius of the cross-section of water stream when it hits the ground is x. Then:

Which of the following option correctly describes the variation of the speed $\upsilon$ and acceleration 'a' of a point mass falling vertically in a viscous medium that applies a force $F=-k\upsilon$ , where 'k' is a constant, on the body? (Graphs are schematic and not drown to scale)

In the following $I$ refers to current and other symbols have their usual meaning. Choose the option that corresponds to the dimensions of electrical conductivity:

A pendulum made of a uniform wire of cross sectional area A has time period T. When an additional mass M is added to its bob, the time period changes to ${T}_{M}$ . If the Young's modulus of the material of the wire is $Y$, then $\frac{1}{Y}$ is equal to: ($g=$gravitational acceleration)

A very long (length $L$) cylindrical galaxy is made of uniformly distributed mass and has radius $R(R<<L)$. A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of the star is $T$ and its distance from the galaxy's axis is $r$, then

Which of the following most closely depicts the correct variation of the gravitation potential, $V(r)$ with distance $r$ due to a large planet of radius $R$ and uniform mass density? (figures are not drawn to scale)

From a solid sphere of mass $M$ and radius $R$, a cube of the maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is:

From the top of a $64$ metres high tower, a stone is thrown upwards vertically with the velocity of $48 m/s.$ The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration $g=32 m/{s}^{2}$, is:

A uniform thin rod AB of length $L$ has linear mass density $\mu (x)=a+\frac{bx}{L}$, where $x$ is measured from A. If the CM of the rod lies at a distance of $(\frac{7}{12}L)$ from A, then $a$ and $b$ are related as:

If electronic charge $e$, electron mass $m$, speed of light in vacuum $c$ and Planck's constant $h$ are taken as fundamental quantities, the permeability of vacuum ${\mu }_{0}$ can be expressed in units of:

If two glass plates have water between them and are separated by very small distance (see figure), it is very difficult to pull them apart. It is because the water in between forms cylindrical surface on the side that gives rise to lower pressure in the water in comparison to atmosphere. If the radius of the cylindrical surface is R and surface tension of water is T then the pressure in water between the plates is lower by: 

A particle of mass $\text{m}$ moving in the $x$ direction with speed $2v$ is hit by another particle of mass $2m$ moving in the $y$ direction with speed $v.$ If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to:

A particle of mass $2\mathrm{kg}$ is on a smooth horizontal table and moves in a circular path of radius $0.6m$. The height of the table from the ground is $0.8m$. If the angular speed of the particle is $12 \mathrm{rad} {s}^{-1}$ , the magnitude of its angular momentum about a point on the ground right under the center of the circle is:

Two stones are thrown up simultaneously from the edge of a cliff $240m$ high with an initial speed of $10m{s}^{-1}$ and $40m{s}^{-1}$ respectively. Which of the following graph best represents the time variation of the relative position of the second stone with respect to the first? (Assume stones do not rebound after hitting the ground and neglect air resistance, take $g=10 {\mathrm{ms}}^{-2}$)(the figure are schematic and not drawn to scale)

A beaker contains a fluid of density $\rho$$\frac{kg}{{m}^{3}}$ , specific heat $S\frac{J}{k{g}^{o}C}$ and viscosity $\eta$ . The beaker is filled up to height h. To estimate the rate of heat transfer per unit area $(\frac{\overset{˙}{Q}}{A})$ by convection when beaker is put on a hot plate, a student proposes that it should depend on $\eta$ , $(\frac{S\Delta \theta }{h})$ and $(\frac{1}{\rho g})$ when $\Delta \theta$ ( ${in}^{o}C$ ) is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for $(\frac{\overset{˙}{Q}}{A})$ is:

A large number $(n)$ of identical beads, each of mass $m$ and radius $r$ are strung on a thin smooth rigid horizontal rod of length $L(L\gg r)$ and are at rest at random positions. The rod is mounted between two rigid supports (see figure). If one of the beads is now given a speed $v$, the average force experienced by each support after a long time is (assume all collisions are elastic):