Varying Mass System

Q.

A uniform chain of length l is placed on the table in such a manner that its l' part is hanging over the edge of table without the chain sliding. If the coefficient of friction between the chain and the table is $\mu$ then find the maximum length of chain l' that can hang without the entire chain slipping.

1. 0

2. $\frac{l}{\mu +1}$

3. $\frac{l}{\mu +1}$

4. $\frac{\mu l}{\mu +1}$

Q. Sand kept inside the trolley drains out from its floor at a constant rate of $0.2\text{kg/s}$. A force $15\text{N}$ is acting on the trolley as shown in figure. After time $5\text{sec}$, which of the following option(s) is/are true?

1. Mass of trolley is $1\text{kg}$
2. Net force on trolley is $15\text{N}$
3. Velocity of trolley is $75\ln 2\text{m/s}$
4. Velocity of trolley is $75\ln 3\text{m/s}$

Q. A machine gun is mounted on a $2000\text{kg}$ car on a horizontal frictionless surface. At some instant the gun fires bullets of mass $10\text{gm}$ with a velocity of $500\text{m/sec}$ with respect to the car. The number of bullets fired per second is ten. The average thrust on the system is

1. $550\text{N}$
2. $50\text{N}$
3. $250\text{N}$
4. $250\text{Dyne}$

Q. A cart loaded with sand moves along a horizontal floor due to constant force $F=10\text{N}$ acting in the direction of the cart's velocity vector. In the process, sand spills through a hole at the bottom at a constant rate, $\mu =5\text{kg/s}$ . Find the velocity of the cart at the time $t=5\text{sec}$, if at the initial moment $t=0$, the cart loaded with sand has mass $m_0=100\text{kg}$ and its velocity was equal to zero. Friction is to be neglected. [Take $\ln \left(\frac{4}{3}\right)=0.288$ and $g=10{\text{m/s}}^2$]

1. $1.428\text{m/s}$
2. $0.823\text{m/s}$
3. $1.1\text{m/s}$
4. $0.576\text{m/s}$

Q. Sand is poured gently at a constant rate of $0.1\text{kg/s}$ on the trolley of initial mass $2\text{kg}$ kept over a smooth surface at rest. The external force acting on the trolley is $10\text{N}$ as shown in the figure. Then, identify the correct statement(s) for the trolley after time $10\text{sec}$:

1. Mass of trolley is $3\text{kg}$
2. Net force on trolley is $\frac{20}{3}\text{N}$
3. Velocity of trolley is $\frac{100}{3}\text{m/s}$
4. Velocity of trolley is $\frac{200}{3}\text{m/s}$

Q. A cart loaded with sand moves along a horizontal floor due to constant force $F=10\text{N}$ acting in the direction of the cart's velocity vector. In the process, sand spills through a hole at the bottom at a constant rate, $\mu =5\text{kg/s}$ . Find the velocity of the cart at the time $t=5\text{sec}$, if at the initial moment $t=0$, the cart loaded with sand has mass $m_0=100\text{kg}$ and its velocity was equal to zero. Friction is to be neglected. [Take $\ln \left(\frac{4}{3}\right)=0.288$ and $g=10{\text{m/s}}^2$]

1. $1.428\text{m/s}$
2. $0.823\text{m/s}$
3. $1.1\text{m/s}$
4. $0.576\text{m/s}$

Q.

A uniform chain of length l is placed on the table in such a manner that its l' part is hanging over the edge of table without the chain sliding. If the coefficient of friction between the chain and the table is $\mu$ then find the maximum length of chain l' that can hang without the entire chain slipping.

1. 0

2. $\frac{l}{\mu +1}$

3. $\frac{l}{\mu +1}$

4. $\frac{\mu l}{\mu +1}$

Q. A tank cart is filled with water as shown in the figure. The initial mass of the cart with water is $m_0=100\text{kg}$. At time $t=0$, a hole is made in the left wall of the cart and water starts spilling out from the cart, with a constant velocity of $v_r=10\text{m/s}$ with respect to the cart. The rate of ejection of water is $r=2\text{kg/s}$. Find the acceleration of the cart after time $t=5\text{s}$.

1. $0.11{\text{m/s}}^2$
2. $0{\text{m/s}}^2$
3. $0.22{\text{m/s}}^2$
4. $0.4{\text{m/s}}^2$

Q. The mass of a rocket is $2400\text{kg}$. The nozzle of the rocket is designed to give velocity of $200\text{m/s}$ to the gases produced after combustion of the fuel. If the rocket has to achieve an acceleration of $15{\text{m/s}}^2$, what should be the rate of burning of the fuel? (Take $g=10{\text{m/s}}^2$)

1. $200\text{kg/s}$
2. $250\text{kg/s}$
3. $400\text{kg/s}$
4. $300\text{kg/s}$

Q. Sand is poured gently at a constant rate of $0.1\text{kg/s}$ on the trolley of initial mass $2\text{kg}$ kept over a smooth surface at rest. The external force acting on the trolley is $10\text{N}$ as shown in the figure. Then, identify the correct statement(s) for the trolley after time $10\text{sec}$:

1. Mass of trolley is $3\text{kg}$
2. Net force on trolley is $\frac{20}{3}\text{N}$
3. Velocity of trolley is $\frac{100}{3}\text{m/s}$
4. Velocity of trolley is $\frac{200}{3}\text{m/s}$

Q. A cart loaded with sand having total mass $900\text{kg}$ moves on a straight horizontal road under the action of a force $60\text{N}$. If cart is starting from rest and sand spills through a small hole at the bottom of cart at a rate of $0.25\text{kg/s},$ what will be the speed of cart after $10$ $\text{minutes}$?
Given: $g=10{\text{m/s}}^2,\ln \left(\frac{5}{6}\right)\approx -0.2$

1. $50\text{m/s}$
2. $68\text{m/s}$
3. $24\text{m/s}$
4. $48\text{m/s}$
   

Q. Sand is being dropped on a conveyer belt at the rate of $Mkg/s$. The force necessary to keep the belt moving with a constant velocity of $vm/s$ will be

1. $\frac{Mv}{2}N$
2. Zero
3. $MvN$
4. $2MvN$

Q. A chain of length $L=10\text{m}$ and mass $m=20\text{kg}$ is allowed to fall on a table such that the part falling on the table comes to rest immediately. The force acting on the table when $5\text{m}$ of it has fallen on the table is [Take $g=10{\text{m/s}}^2$]

1. $300\text{N}$
2. $600\text{N}$
3. $200\text{N}$
4. $100\text{N}$

Q. As shown in the figure, sand is falling from a stationary hopper onto a freight car which is moving with a uniform velocity $v_0=10\text{m/s}.$ The sand falls at the rate of $\mu =5\text{kg/s}.$ How much force is needed to keep the freight car moving at the speed $v_0$?

1. $50\text{N}$
2. $0\text{N}$
3. $25\text{N}$
4. $20\text{N}$

Q. A machine gun is mounted on a $2000\text{kg}$ car on a horizontal frictionless surface. At some instant the gun fires bullets of mass $10\text{gm}$ with a velocity of $500\text{m/sec}$ with respect to the car. The number of bullets fired per second is ten. The average thrust on the system is

1. $550\text{N}$
2. $50\text{N}$
3. $250\text{N}$
4. $250\text{Dyne}$

Q. A flatcar of mass $m_0=20\text{kg}$ is moving to the right due to a constant horizontal force $F=10\text{N}$. Sand spills on the flatcar from a stationary hopper. The velocity of loading is constant and equal to $\mu =1\text{kg/s}$.



Find the acceleration of the flatcar after $t=20\text{s}$.

1. $\frac{1}{8}{\text{m/s}}^2$
2. $\frac{1}{40}{\text{m/s}}^2$
3. $0$
4. $5{\text{m/s}}^2$

Q. A truck is moving with an acceleration of $2{\text{m/s}}^2$ due to a constant horizontal force $F$. Suddenly, a hail storm starts with a constant rate of mass $\mu =20\text{kg/s}$ getting deposited on the truck. Find the force needed to maintain the acceleration of the truck after $2$ seconds. Take $g=10{\text{m/s}}^2.$ Speed of the hailstorm is $2\text{m/s}$ (perpendicular to the velocity of the truck) and mass of the truck is initially $m_0=50\text{kg}$.

1. $324\text{N}$
2. $0\text{N}$
3. $50\text{N}$
4. $300\text{N}$

Q. A cart of total mass $50\text{kg}$ is at rest on a horizontal road having coefficient of friction 0.1. Gases are ejected from this cart backwards with velocity $20\text{m/s}$ w.r.t the cart. The rate of ejection of gases is $2\text{kg/s}$. The cart will start moving after time $t=$ (seconds) [Take $g=10{\text{m/s}}^2$]

Q. A cart of total mass $50\text{kg}$ is at rest on a horizontal road having coefficient of friction 0.1. Gases are ejected from this cart backwards with velocity $20\text{m/s}$ w.r.t the cart. The rate of ejection of gases is $2\text{kg/s}$. The cart will start moving after time :
[Take $g=10{\text{m/s}}^2$]

1. $t=2\text{s}$
2. $t=3\text{s}$
3. $t=5\text{s}$
4. $t=10\text{s}$