A train is moving with velocity 20 m/sec. on this, dust is falling at the rate of 50 kg/min. The extra force required to move this train with constant velocity will be

To find the extra force required to keep the train moving at a constant velocity while dust is falling on it, we can use the concept of variable mass dynamics derived from Newton's second law of motion.

Newton's second law states that force is the rate of change of momentum:
$F=\frac{dp}{dt}$
Since momentum $p=mv$, where $m$ is the mass and $v$ is the velocity, the rate of change of momentum is given by:
$F=\frac{d}{dt}\left(mv\right)=v\frac{dm}{dt}+m\frac{dv}{dt}$

The question specifies that the train must move with a constant velocity. Therefore, the acceleration is zero:
$\frac{dv}{dt}=0$
Substituting this back into the force equation gives the required extra force:
$F=v\frac{dm}{dt}$

Now, we identify the given values from the problem:
1. Constant velocity of the train, $v=20\text{m/sec}$
2. Rate at which dust is falling, $\frac{dm}{dt}=50\text{kg/min}$

Next, we convert the rate of mass accumulation to the standard SI unit (kilograms per second):
Since $1\text{minute}=60\text{seconds}$, we have:
$\frac{dm}{dt}=\frac{50}{60}\text{kg/sec}=\frac{5}{6}\text{kg/sec}$

Finally, we substitute the values of $v$ and $\frac{dm}{dt}$ into the force formula:
$F=20\times \frac{5}{6}\text{N}$
$F=\frac{100}{6}\text{N}$
$F\approx 16.66\text{N}$

Thus, the extra force required to move the train with a constant velocity is approximately 16.66 N.