Velocity of COM

Q. Let a total charge $2Q$ be distributed in a sphere of radius $R$, with the charge density given by $\rho \left(r\right)=kr$, where $r$ is the distance from the centre. Two charges $A$ and $B$, of $-Q$ each, are placed on diametrically opposite points, at equal distance $a$, from the centre. If $A$ and $B$ do not experience any force, then

1. $a=2^{\left(\frac{-1}{4}\right)}R$
2. $a=\frac{R}{\sqrt{3}}$
3. $a=8^{\left(\frac{-1}{4}\right)}R$
4. $a=\frac{4}{2^{\left(\frac{1}{4}\right)}}$

Q. Two particles which are initially at rest, move towards each due to mutual attraction. If their speeds are $v$ and $2v$ at any instant, then the speed of centre of mass of the system will be

1. $2v$
2. $v$
3. $1.5v$
4. $0$

Q. The boxes of masses $2\text{kg}$ and $8\text{kg}$ are connected by a massless string passing over smooth pulleys. Calculate the time taken by box of mass $8\text{kg}$ to strike the ground starting from rest.
$(\text{Use}g=10{\text{m/s}}^2)$ :

1. $0.2\text{s}$
2. $0.34\text{s}$
3. $0.25\text{s}$
4. $0.4\text{s}$

Q.

Is Friction independent of the actual area of contact?

Q. A man $(\text{mass}=50\text{kg})$ and his son $(\text{mass}=20\text{kg})$ are standing on a frictionless surface facing each other. The man pushes his son, so that he starts moving at a speed of $0.70{\text{ms}}^{-1}$ with respect to the man. The speed of the man with respect to the surface is

1. $0.14{\text{ms}}^{-1}$
2. $0.20{\text{ms}}^{-1}$
3. $0.47{\text{ms}}^{-1}$
4. $0.28{\text{ms}}^{-1}$

Q.

What is the time rate of change of momentum called?

Q.

The mass of a hydrogen molecule is$3.32\times {10}^{-27}\mathrm{kg}$. If ${10}^{23}$ hydrogen molecules strike, per second, a fixed wall of area $2{\mathrm{cm}}^2$ at an angle of $45°$ to the normal, and rebound elastically with a speed of ${10}^3\mathrm{m}/\mathrm{s}$, then the pressure on the wall is nearly:

Q. When a bullet is fired from a gun, which has more momentum? Bullet or gun?

Q.

Two blocks of masses $m_1$ and $m_2$ are connected by a spring of constant k (given figure).The block of mass $m_2$ is given a sharp impulse so that it acquires a velocity $v_0$ towards right.Find (a) the velocity of the centre of mass, (b) the maximum elongation that the spring will suffer.

Q.

How can the average density of the earth can be determined?

Q. The linear charge density of a thin rod of length $L$ is given by $\lambda =\left({\lambda }_{0}x\right)\text{Coulomb/m},[0\le x<\frac{L}{2}]$ and $\lambda =\left({\lambda }_0{x}^2\right)\text{Coulomb/m},[\frac{L}{2}\le x\le L]$. Find the total charge on rod. $(x$ is measure from one end of the rod$)$.

1. $\frac{{\lambda }_0{L}^2}{24}(3+7L)$
2. $\frac{{\lambda }_0{L}^3}{8}(1+5L)$
3. $\frac{{\lambda }_0{L}^2}{4}(1+7L)$
4. ${\lambda }_0{L}^2(1+7L)$

Q. Two particles of mass $1\text{kg}$ and $3\text{kg}$ move towards each other under their mutual force of attraction. No other forces act on them. When the velocity of approach of the two particles is $2\text{m/s}$, their centre of mass has a velocity of $0.5\text{m/s}$. When the velocity of approach becomes $3\text{m/s}$, the velocity of the centre of mass of the system of particles is

1. $0.75\text{m/s}$
2. $1.25\text{m/s}$
3. $1.5\text{m/s}$
4. $0.5\text{m/s}$

Q.

A $16kg$ block moving on a frictionless horizontal surface with a velocity of $4m{s}^{-1}$compresses an ideal spring and comes to rest. If the force constant of the spring be $100N{m}^{-1}$, then the spring is compressed by

1. $1.6m$

2. $4m$

3. $6.1m$

4. $3.2m$

Q. The linear charge density of a thin rod of length $l$ varies with the distance $x$ from the left-most end as $\lambda ={\lambda }_0{x}^2(0\le x\le l)$. The total charge on the rod is

1. $\frac{{\lambda }_0{l}^3}{3}$
2. $\frac{{\lambda }_0{l}^3}{4}$
3. $\frac{2{\lambda }_0{l}^2}{3}$
4. $\frac{{\lambda }_0{l}^2}{3}$

Q.

Consider a spherical gaseous cloud of mass density $\rho \left(r\right)$ in a free space where $r$ is the radial distance from its centre. The gaseous cloud is made of particles of equal mass $m$ moving in circular orbits about their common centre with the same kinetic energy $K$. The force acting on the particles is their mutual gravitational force. If $\rho \left(r\right)$ is constant with time. The particle number density $n\left(r\right)=\frac{\rho \left(r\right)}{m}$ is:

($g=$universal gravitational constant)

1. $\frac{3K}{\mathrm{\pi }{r}^2{\mathrm{m}}^2\mathrm{G}}$

2. $\frac{K}{2\mathrm{\pi }{r}^2{\mathrm{m}}^2\mathrm{G}}$

3. $\frac{K}{\mathrm{\pi }{r}^2{\mathrm{m}}^2\mathrm{G}}$

4. $\frac{K}{6\mathrm{\pi }{r}^2{\mathrm{m}}^2\mathrm{G}}$

Q. It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is $p_d$, while for its similar collision with carbon nucleus at rest, fractional loss of energy is $p_c$
The values of $p_d$ and $p_c$ are respectively :

1. $(0,0)$
2. $(0,1)$
3. $(0.89,0.28)$
4. $(0.28,0.89)$

Q. A sphere of radius $R$ has charge density per unit volume, $\rho \left(r\right)=A(R-r),$ for $0<r<R$ ( Here $A$ is constant). If total electric charge inside the sphere is $Q$, then the value of $A$ in terms of $Q$ and $R$ is

1. $\frac{2Q}{\pi R^4}$
2. $\frac{3Q}{\pi R^4}$
3. $\frac{3Q}{2\pi R^4}$
4. $\frac{3Q}{\pi R}$

Q. If the external forces acting on a system have zero resultant, the center of mass

1. must not move
2. must not accelerate
3. may move
4. may accelerate

Q. Two particles of equal masses have velocities ${\overrightarrow{v}}_1=\left(2\hat{i}\right)\text{m/s}$ and ${\overrightarrow{v}}_2=\left(2\hat{j}\right)\text{m/s}$. The first particle has an acceleration of ${\overrightarrow{a}}_1=(3\hat{i}+3\hat{j}){\text{m/s}}^2$ while the acceleration of the other particle is zero. The path of the centre of mass of the two particles system is a/an

1. parabola
2. straight line
3. ellipse
4. circle

Q.

Two particles A and B of masses 1 kg and 2 kg respectively are kept 1 m apart and are released to move under mutual attraction. Find the speed of A when that of B is 3 cm/hour. What is the separation between the particles at this instant?

1. v = 2 × 10^-5m/s , d = 0.81m

2. v = 2 × 10^-5m/s , d = 0.31m.

3. v = 10^-5m/s , d = 0.31m.

4. v = 10^-5m/s , d = 0.41m.

Q. A stone is dropped into water from a bridge 44.1 m above the water. Another stone is thrown vertically downward 1 sec later. Both strike the water simultaneously. What was the initial speed of the second stone

1. 17.15 m/s
2. 12.25 m/s
3. 16.23 m/s
4. 14.75 m/s

Q. A man of mass $50\text{kg}$ is standing on a platform of mass $200\text{kg}$ kept on a smooth horizontal surface. The man starts moving on the platform with a velocity $20\text{km/hr}$ relative to the platform. Find the recoil speed of the platform.

1. $4\text{km/hr}$
2. $2\text{km/hr}$
3. $3\text{km/hr}$
4. $1\text{km/hr}$

Q. When a particle executes $\text{SHM}$, the nature of graphical representation of velocity as a function of displacement is:

1. elliptical
2. parabolic
3. straight line
4. circular

Q.

Akhtar, Kiran, and Rahul were riding in a motorcar that was moving with a high velocity on an expressway when an insect hit the windshield and got stuck on the windscreen. Akhtar and Kiran started pondering over the situation. Kiran suggested that the insect suffered a greater change in momentum as compared to the change in momentum of the motorcar (because the change in the velocity of the insect was much more than that of the motorcar). Akhtar said that since the motorcar was moving with a larger velocity, it exerted a larger force on the insect. And as a result, the insect died. Rahul while putting an entirely new explanation said that both the motorcar and the insect experienced the same force and a change in their momentum. Comment on these suggestions.

Q. A satellite of mass $m$ is revolving around the earth in a circular orbit of certain radius . At certain point of its orbit , the direction of motion of satellite is suddenly changed by angle $\alpha (Givencos\alpha =\frac{3}{5})$ by turing its velocity vector , such that speed remains constant . The satellite, consequently goes to elliptical orbit around earth . Find ratio of speed at apogee to speed at perigee.

1. $\frac{1}{3}$
2. 9
3. $\frac{1}{9}$
4. 3

Q. A satellite is launched in the equatorial plane is such a way that it can transmit signals up to ${60}^{\circ }$ latitude on the earth. Then the angular velocity of the satellite is

1. $\sqrt{\frac{GM}{8R^3}}$
2. $\sqrt{\frac{GM}{2R^3}}$
3. $\sqrt{\frac{GM}{4R^3}}$
4. $\sqrt{\frac{3GM}{8R^3}}$

Q.

The acceleration due to gravity near the surface of a planet of radius $R$ and density $d$ is proportional to

1. $\frac{d}{R^2}$

2. $d{R}^2$

3. $dR$

4. $\frac{d}{R}$

Q. Four particles of masses $1\text{kg},2\text{kg},3\text{kg}$ and $4\text{kg}$ are placed at the four vertices $\text{A, B, C}$ and $\text{D}$ of a square of side $1\text{m}$ as shown in figure. Find the position of centre of mass of the particles.

1. $(0.3,0.5)\text{m}$
2. $(0.1,0.2)\text{m}$
3. $(0.5,0.3)\text{m}$
4. $(0.5,0.5)\text{m}$

Q.

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centers. The moment of inertia and angular velocity of the first disc are $0.1kg/m^2$ and $10rad/s$ respectively while those for the second one are $0.2kg/m^2$ and $5rad/s$ respectively. At some instant, they get stuck together and start rotating as a single system about their common axis with some angular speed. The Kinetic energy of the combined system is:

1. $\frac{2}{3}J$

2. $\frac{10}{3}J$

3. $\frac{5}{3}J$

4. $\frac{20}{3}J$

Q.

A ball is thrown vertically upward with a velocity of $60m/s$. The maximum height to which the ball rises is