Varying Mass System

Q.

Prove that $F=ma.$

Q.

State Newtons second law of motion. Use this law to find the method to measure forces acting on an object.

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

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

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. 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. 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 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. 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 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. $t=10\text{s}$

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

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 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 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. $GSLV$ launched a rocket with a ejection speed of $200\text{m/s}$ and with $20{\text{m/s}}^2$ acceleration. Mass of the rocket is $2000\text{kg}$. What will be the rate of consumption of the fuel? (Take $g=10{\text{m/s}}^2$)

1. $400\text{kg/s}$
2. $300\text{kg/s}$
3. $500\text{kg/s}$
4. $600\text{kg/s}$

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. $16.1\text{m/s}$