Free Fall

Q.

The relation between time $t$ and distance $x$ for a moving body is given as $t=m{x}^2+nx$, where $m$ and $n$ are constants. The retardation of the motion is :
$\left(\text{Where}v\text{stands for velocity}\right)$

Q. Two spherical balls having equal masses with radius of $5\text{cm}$ each are thrown upwards along the same vertical direction at an interval of $3\text{s}$ with the same initial velocity of $35\text{m/s}$, then these balls will collide at a height of ____ $\text{m}$.
(Take $g=10{\text{m/s}}^2$)

Q.

The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity

1. will be directed towards the centre but not the same everywhere.

2. cannot be zero at any point.

3. will have the same value everywhere but not directed towards the centre.

4. will be same everywhere in magnitude directed towards the centre.

Q. Efficiency of a cyclic process is $50\%$ and heat interactions are $Q_1=+1915\text{J},Q_2=-40\text{J},Q_3=+125\text{J}$ and $Q_4=-Q\text{J}$. Find the value of $Q$.

1. $1080\text{J}$
2. $1980\text{J}$
3. $-1080\text{J}$
4. $980\text{J}$

Q.

The acceleration of the projectile motion is zero when it is at the topmost point.

1. True
2. False

Q. A body of weight $72\text{N}$ moves from the surface of earth to a height equal to half the radius of earth. Gravitational force will be

1. $36\text{N}$
2. $32\text{N}$
3. $144\text{N}$
4. $50\text{N}$

Q.

From a cliff of $49m$ high, a man drops a stone. One second later, he throws another stone. They both hit the ground at the same time. Find out the speed with which he threw the second stone.

Q.

A ball is dropped from a bridge at a height of $176.4\mathrm{m}$ over a river. After $2\mathrm{s}$, a second ball is thrown straight downwards. What should be the initial velocity of the second ball so that both hit the water simultaneously?

1. $2.45{\mathrm{ms}}^{-1}$

2. $49{\mathrm{ms}}^{-1}$

3. $25.7{\mathrm{ms}}^{-1}$

4. $24.5{\mathrm{ms}}^{-1}$

Q. In the figure shown neglecting friction andd mass of pulleys, what is the aacceleration of mass $B$?

1. $\frac{g}{3}$
2. $\frac{5g}{2}$
3. $\frac{2g}{5}$
4. $\frac{2g}{3}$

Q. Two identical thin rings each of radius $R$ are coaxially placed at a separation of $R$. If the rings have a uniform mass distribution and masses $m_1$ and $m_2$ respectively, then the work done in moving a mass $m$ from centre of the ring having mass $m_1$ to center of the other ring is

1. $0$
2. $\frac{Gm(2-\sqrt{2})(m_1-m_2)}{2R}$
3. $\frac{Gm(2-\sqrt{2})(m_2-m_1)}{2R}$
4. $\frac{Gm(2-\sqrt{2})(m_1+m_2)}{2R}$
   

Q.

An astronaut inside a small spaceship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size, can he hope to detect gravity?

Q. In the arrangement shown in the figure, all the surfaces are frictionless. The masses of the blocks are $m_1=20kg$ and $m_2=30kg$. If $F=180N$, then the ratio of acceleration of mass $m_2$ to that of $m_1$ will be

Q. When a mass $2\text{kg}$ is shifted from infinity to a point in gravitational field, it is found that work done by external force is $-100\text{J}$. If the mass is doubled and taken again from the same point to infinity without accelerating it, then the work done by gravitational force will be

1. $-100\text{J}$
2. $-200\text{J}$
3. $200\text{J}$
4. $400\text{J}$

Q. A fixed sphere of radius $R$ and uniform density ρ has a spherical cavity of radius $R/2$ such that the surface of the cavity passes through the centre of the sphere. $A$ particle of mass $m$ is located at the centre $\left(A\right)$ of the cavity. The gravitational field at A is $\frac{2}{3}\pi \rho GR$. Calculate the velocity with which the particle strikes the centre $O$ of the sphere. (Neglect earth's gravity)

1. ${\left(\frac{2\pi G\rho R^2}{6}\right)}^{\frac{1}{2}}$
2. ${\left(\frac{2\pi G\rho R^2}{3}\right)}^{\frac{1}{2}}$
3. ${\left(\frac{2\pi G\rho R^2}{9}\right)}^{\frac{1}{2}}$
   
4. None of these

Q. Write down the average value of $g$ on the earth surface.

Q. A coin and a feather are dropped together in a vacuum, then -

1. the coin will reach the ground first.
2. the feather will reach the ground first.
3. both the bodies will reach the ground together.
4. the feather will not fall down.

Q.

When a rifle is fired at a distant target, the barrel is not exactly lined up on the target. why? Does the angle of correction depends upon the distance of targets ?

Q. A lift is falling with an acceleration \(2~m/s^2\). A ball of mass \(100~gm\) is attached at one end of the string and the other end is fixed to the ceiling of the lift. The ball remains at rest relative to lift. The tension (in \(N\)) in the string is: \((g=10~ m/s^2\) )

Q. Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and
momentum is plotted along vertical axis. The phase space diagram is $x\left(t\right)vs.p\left(t\right)$ curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

1. D
2. B
3. C
4. A

Q. A planet having a same density as that of earth but its radius is $4$ times bigger than the radius of the earth. If the acceleration due to gravity on the surface of earth is $g$ and that on the surface of the planet is $g^{\prime }$, then

1. $g^{\prime }=\frac{g}{4}$
2. $g^{\prime }=16g$
3. $g^{\prime }=9g$
4. $g^{\prime }=4g$

Q. A ball is dropped from the certain height on the surface of glass. It collides elastically and comes back to initial position. If this process is repeated then the corresponding velocity- time graph is

1. 
2. 
3. 
4. 

Q. Assuming that the earth has constant density, at what distance $\left(d\text{in km}\right)$ from the earth's surface the gravity above the earth is equal to that below the surface.

Q. In the figure shown blocks $A$ and $B$ are kept on a wedge $C.A,B$ and $C$ each have mass $m$. All surfaces are smooth. Find the acceleration (in $m/s^2$) of $C$.

Q. Two stars of masses $3\times {10}^{31}kg$ each, and at distance $2\times {10}^{11}m$ rotate in a plane about their common centre of mass $O$. A meteorite passes through $O$ moving, perpendicular to the star's rotation plane. In order to escape from the gravitational field of this double star, the minimum speed that meteorite should have at $O$ is (take gravitational constant $G=6.67\times {10}^{(-11)}N{m}^{2}k{g}^{(-2)})$

1. $24\times {10}^{4}m/s$
2. $3.8\times {10}^{4}m/s$
3. $1.4\times {10}^{5}m/s$
4. $2.8\times {10}^{5}m/s$

Q. A ring of mass $M$ and radius $R$ is released on an inclined plane as shown. If the coefficient of friction $\mu <\frac{1}{2}\tan \theta$, then during a displacement $l$
Diagram

1. Work done by the force of friction $=mgl(\sin \theta -\mu \cos \theta )$
2. Work done by the force of friction is zero
3. Acceleration of the ring is zero
4. Acceleration of the ring $=g\sin \theta -\mu g\cos \theta$

Q. A force of $100N$ is applied on a block of mass $3kg$ as shown in the figure. THe coefficient of the friction between the surface and the block is $\mu =\frac{1}{\sqrt{3}}$. The frictional force acting on the block is

1. $10m/s^2\text{upwards}$
2. $10m/s^2\text{downward}$
3. $\text{zero}$
4. $\text{none of these}$

Q. A small body is initially at a distance $r$ from the centre of earth. $r$ is greater than the radius of the earth. If it takes $W$ $\text{joules}$ of work to move the body from this position to another position at a distance $2r$ measured from the centre of earth, then how many $\text{joules}$ would be required to move it from this position to a new position at a distance of $3r$ from the centre of the earth.

1. $\frac{W}{5}$
2. $\frac{W}{3}$
3. $\frac{W}{2}$
4. $\frac{W}{6}$

Q.

There have been suggestions that the value of the gravitational constant G becomes smaller when considered over very large time period (in billions of years) in the future. If that happens, for our earth,

1. we will become hotter after billions of years.

2. Nothing will change.

3. we will be going around but not strictly in closed orbits.

4. after sufficiently long time we will leave the solar system

Q. A particle of mass $‘m^{\prime }$ oscillates along the horizontal diameter $AB$ inside a smooth spherical shell of radius $R$. At any instant $K.E.$ of the particle is $K$. Then force applied by particle on the shell at this instant is:

1. $\frac{K}{2R}$
2. $\frac{3K}{R}$
3. $\frac{2K}{R}$
4. $\frac{K}{R}$

Q. A string of negligible mass going over a clamped pulley of mass $m$ supports a block of mass $M$ as shown in the figure. The force on the pulley by the clamp is given by

1. $\sqrt{(M+m{)}^2+M^{2}g}$
2. $\sqrt{2}Mg$
3. $\sqrt{2}mg$
4. $\sqrt{(M+m{)}^2+m^{2}g}$