JEE Main 2016 — Mechanics

A bottle has an opening of radius $a$ and length $b$. A cork of length $b$ and radius $(a+\Delta a)$ where $(\Delta a\ll a)$, is compressed to fit into the opening completely (see figure). If the bulk modulus of cork is $B$ and the coefficient of friction between the bottle and cork is $\mu$, then the force needed to push the cork into the bottle is 

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 cubical block of side $30\mathrm{cm}$ is moving with velocity $2m{s}^{-1}$ on a smooth horizontal surface. The surface has a bump at a point $O$ as shown in the figure. The angular velocity (in rad/s) of the block immediately after it hits the bump, is : 

A particle of mass $m$ is acted upon by a force $F$ given by the empirical law $F=\frac{R}{{t}^{2}} v(t).$ If this law is to be tested experimentally by observing the motion starting from rest, the best way is to plot

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 particle of mass M is moving in a circle of fixed radius R in such a way that its centripetal acceleration at time t is given by ${n}^{2}R{t}^{2}$, where $n$ is a constant. The power delivered to the particle by the force acting on it, is :

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 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 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 roller is made by joining together two cones at their vertices O. It is kept on two rails AB and CD which are placed asymmetrically (see figure), with its axis perpendicular to CD and its centre O at the centre of line joining AB and CD (see figure). It is given a light push so that it starts rolling with its centre O moving parallel to CD in the direction shown. As it moves, the roller will tend to: 

A satellite is revolving in a circular orbit at a height $h$ from the earth's surface (radius of earth $R$; $\text{h} \text{<<} \text{R}$ ). The minimum increase in its orbital velocity required, so that the satellite could escape from the earth's gravitational field, is close to (Neglect the effect of atmosphere.)

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?

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 thin $1m$ long rod has a radius of $5\mathrm{mm}$. A force of $50{\text{π×10}}^{3}N$ is applied at one end to determine its Young's modulus. Assume that the force is exactly known. If the least count in the measurement of all lengths is $0.01\mathrm{mm}$, which of the following statements is false?

A uniformly tapering conical wire is made from a material of Young's modulus $Y$ and has a normal, unextended length $L$. The radii, at the upper and lower ends of this conical wire, have values $R$ and $3R$, respectively. The upper end of the wire is fixed to a rigid support and a mass $M$ is suspended from its lower end. The equilibrium extended length, of this wire, would equal:

An astronaut of mass $m$ is working on a satellite orbiting the earth at a distance $h$ from the earth's surface. The radius of the earth is $R$, while its mass is $M$. The gravitational pull ${F}_{G}$ on the astronaut is

$A, B, C$, and $D$ are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation $AD=C ln( BD )$ holds true. Then which of the combination is not a meaningful quantity?

Concrete mixture is made by mixing cement, stone and sand in a rotating cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute (rpm) to ensure proper mixing is close to : (Take the radius of the drum to be 1.25 m and its axle to be horizontal) :

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 

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:

 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:

The figure shows an elliptical path $ABCD$ of a planet around the sun $S$ such that the area of triangle $CSA$ is $\frac{1}{4}$ the area of the ellipse. (see figure) with $DB$ as the major axis, and $CA$ as the minor axis. If ${t}_{1}$ is the time taken for the planet to go over path $ABC$ and ${t}_{2}$ for path taken over $CDA$ then: 

The velocity-time graph of a particle of mass $10\mathrm{kg}$ is shown in the figure. The net work done on the particle in the first two seconds of the motion is 

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)