Varying Mass System

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

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

1. 250 dyne
2. 550 N
3. 50 N
4. 250 N

Q. A uniform metal chain is placed on a rough table such that one end of chain hangs down over the edge of the table. When one-third of its length hangs down over the edge, the chain starts sliding. Then, the value of coefficient of static friction is-

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

Q.

Which law gives a quantitative measurement of the force?

Q.

A particle moves with a uniform velocity. Which of the following statements about the motion of particle is correct?

1. Its acceleration is zero

2. Its acceleration is opposite to the velocity
3. Its speed may be variaable
4. Its speed is zero

Q. The viscous force on a moving block is directly proportional to velocity. The unit of constant of proportionality is

1. $kgs$
2. $kgm{s}^{-3}$
3. $kgm{s}^2$
4. $kg{s}^{-1}$

Q. A plate of mass $M$ is moved with constant velocity $v$ against dust particles moving with velocity $u$ in opposite direction as shown. The density of the dust is $\rho$ and plate area is $A$. Find the force $F$ required to keep the plate moving with uniform velocity.

1. $\rho A(u+v)$
2. $\rho (u+v{)}^2$
3. $\rho A{u}^2$
4. $\rho A(u+v{)}^2$

Q. A cart of mass $M_0$ is moving with velocity $v_0$. At $t=0$, water starts pouring into the cart from a container above the cart at the rate of $\lambda \text{kg/sec}$. Find the velocity of the cart as a function of time.

1. $\frac{M_0{v}_0}{M_0+\lambda t}$
2. None of these
3. $\frac{M_0{v}_0}{M_0-\lambda t}$
4. $\frac{M_0{v}_0}{M_0+2\lambda t}$

Q. A rocket has a mass of $1000\text{kg}$ with $50,000\text{kg}$ of fuel being stored on this rocket. The nozzle of the rocket lets out the exhaust gases at a speed of $500\text{m/s}$. If the rocket consumes fuel at the rate of $100\text{kg/s}$, what is the acceleration of the rocket after $5\text{minutes}$ in a gravity free space?

1. $10{\text{m/s}}^2$
2. $3.95{\text{m/s}}^2$
3. $2.38{\text{m/s}}^2$
4. $1.56{\text{m/s}}^2$

Q.

The acceleration of an object is inversely proportional to _____________

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 rocket is launched with gas ejection speed of $400\text{m/s}$ and acceleration of $25{\text{m/s}}^2$. If mass of the rocket is $4000\text{kg}$, what will be the rate of consumption of the fuel?
(Take $g=10{\text{m/s}}^2$)

1. $250\text{kg/s}$
2. $300\text{kg/s}$
3. $400\text{kg/s}$
4. $350\text{kg/s}$

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

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 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. $0.576\text{m/s}$
4. $1.1\text{m/s}$

Q. If a chain is lowered at a constant speed of $v=1.2\text{m/s}$, determine the normal reaction exerted on the floor as a function of time. The chain has a mass of $80\text{kg}$ and a total length of $6\text{m}$. [Take $g=10{\text{m/s}}^2$]

1. $19.2\text{N}$
2. $(160t+19.2)\text{N}$
3. $(160t+16)\text{N}$
4. $160t\text{N}$

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 car at rest on a horizontal surface (with coefficient of friction 0.1) has total mass $50\text{kg}$. Gases are ejected from this car backwards with relative velocity $20\text{m/s}$. The rate of ejection of gases is $2\text{kg/s}$. Total mass of gas is $20\text{kg}$. Find the maximum speed of the car in $\left(\text{m/s}\right)$
Take $g=10{\text{m/s}}^2,\text{ln}\left(\frac{4}{3}\right)=0.29$

1. $0.8\text{m/s}$
2. $0.2\text{m/s}$
3. $0.4\text{m/s}$
4. $0.6\text{m/s}$

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

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

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

Q. A chain of mass $10kg$ and length $8m$ is resting on a rough horizontal surface $(\mu =0.2)$. A force $F=15N$ is applied as shown. Find the length (in $m$) of the chain for which no friction force acts.

1. $2$
2. $6$
3. $8$
4. $0$

Q. A spaceship of mass $m_0=500\text{kg}$ moves in the absence of an external force with a constant velocity $v_0=50\text{m/s}$. To change the direction of motion, a jet engine is switched on. It starts ejecting a gas jet with velocity $u=10\text{m/s}$ which is constant relative to the spaceship and directed at right angles to the spaceship motion. The engine is shut down when the mass of the spaceship decreases to $400\text{kg}$. Through what angle $\alpha$ does the direction of motion of the spaceship deviate due to the jet engine operation? (Take $\ln \left(1.25\right)=0.223$)

1. $0.45$
2. $0.06$
3. $0.08$
4. $0.045$

Q.

Two particles are travelling due east with different velocities. However, after $4s$ they have the same velocities. During this $4s$ interval, average acceleration of $1^{st}$ particle is $2m{s}^{-2}$ due east while that of $2^{nd}$ particle is $4m{s}^{-2}$ due east. Determine the difference in their speeds at the beginning of $4s$ duration and also fine which one is moving faster initially.

Q. A cart filled with sand of mass $m_0=50\text{kg}$ is moving with constant velocity because of a constant force $F=10\text{N}$ acting in the direction of the cart's velocity vector. Sand starts spilling through a hole in the bottom with a constant velocity at the rate of $\mu =10\text{kg/s}$.. Find the velocity of the cart at the time $t=4\text{s}$. Friction is to be neglected. [Take $\ln \left(5\right)=1.61$]

1. $2.7\text{m/s}$
2. $1.61\text{m/s}$
3. $3.2\text{m/s}$
4. $16.1\text{m/s}$

Q. A trolley of initial mass $m_0$ is kept over a smooth surface, as shown in the figure. A constant force $F$ is applied on it. Sand kept inside the trolley drains out from its floor at a constant rate of $\mu \text{kg/s}$. After time, $t$ find the velocity of trolley.

1. $\frac{F}{\mu }\ln \left(\frac{m_0-\mu t}{m_0}\right)$
2. $\frac{F}{\mu }\ln \left(\frac{m_0}{m_0-\mu t}\right)$
3. $\frac{F}{2\mu }\ln \left(\frac{m_0-\mu t}{m_0}\right)$
4. $\frac{F}{2\mu }\ln \left(\frac{m_0}{m_0-\mu t}\right)$

Q. A block of mass $0.9kg$ attached to a spring of force constant $k$ is lying on a frictionless floor. The spring is compressed to $\sqrt{2}cm$ and the block is at a distance $1/\sqrt{2}cm$ from the wall as shown in the figure. When the block is released, it makes elastic collision with the wall and its period of motion is $0.2sec$. Find the approximate value of $k$.