Question

A box is put on a scale which is adjusted to read zero,when the box is empty. A stream of pebbles is then poured into the box from a height h above its bottom at a rate of n pebbles collide with the box such that they immediately come to rest after collision, the scale reading at time t after the pebbles begin to fill the box is :

• A. $mn\left\{\sqrt{\left(2gh\right)}+gt\right\}$
• B. $\left\{\sqrt{\left(2gh\right)+gt}\right\}$
• C. $\left\{\sqrt{\left(2gh\right)-gt}\right\}$
• D. $mn\left\{\left(2gh\right)-gt\right\}$

Answer 1

The correct option is A $mn\left\{\sqrt{\left(2gh\right)}+gt\right\}$

Given rate of pebbles =n,

thus number of pebbles in the box after time ${}^{\prime }{t}^{\prime }=nt$----(1)

and weight of the pebble in the box after time t $=mgnt$------(2)

where m= mass,g= acceleration due to gravity.

Also given height of fall=h

Now from equation of motion we have,

$h=ut+\frac{1}{2}g{t}^2$ as initial velocity of the pebble $=0m/s$

$h=\frac{1}{2}g{t}^2$

$t=\sqrt{\frac{2h}{g}}$---------(3)

Also we know,

$v=u+gt=gt=\sqrt{2gh}$------(4)

Momentum of pebble $P=mv=m\sqrt{2gh}$-------(5)

As per question pebbles strike at the rate of n, then force required to stop the pebble will be

$F=nP=mn\sqrt{2gh}$--------(6)

Thus the reading of the scale at time t after the pebbles begin to fill the box will be

$W=mgnt+F=mgnt+mn\sqrt{2gh}=mn(gt+\sqrt{2gh})$