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 base from a height h above its bottom at a rate of n pebbles/s. Each pebble has a mass m. If the pebbles collide with the box such that they immediately come to rest after collision, then the scale reading at time t after the pebbles begin to fill the box is [ neglect piling up of pebbles]

Options

A. mnt
B. `mn [sqrt(2gh)+ g t]`
C. mngt
D. zero

Answer

To solve the problem step by step, we will analyze the situation involving the box, the pebbles, and the scale.

### Step 1: Understand the Setup
- We have a box placed on a scale that reads zero when the box is empty.
- Pebbles are dropped into the box from a height \( h \) at a rate of \( n \) pebbles per second.
- Each pebble has a mass \( m \).

### Step 2: Calculate the Mass of Pebbles Added Over Time
- In time \( t \), the total number of pebbles that fall into the box is given by:
\[ N = n \cdot t \]
- The total mass of these pebbles is:
\[ M = N \cdot m = n \cdot t \cdot m \]

### Step 3: Determine the Weight of the Pebbles
- The weight of the pebbles added to the box is given by:
\[ W = M \cdot g = (n \cdot t \cdot m) \cdot g \]
where \( g \) is the acceleration due to gravity.

### Step 4: Calculate the Velocity of the Pebbles Just Before Impact
- The velocity of each pebble just before it hits the bottom of the box can be calculated using the equation of motion:
\[ v = \sqrt{2gh} \]

### Step 5: Calculate the Change in Momentum
- The change in momentum for each pebble when it comes to rest upon hitting the box is given by:
\[ \Delta p = m \cdot v = m \cdot \sqrt{2gh} \]
- Since \( n \) pebbles fall per second, the total change in momentum per second is:
\[ \Delta P = n \cdot m \cdot \sqrt{2gh} \]

### Step 6: Total Force on the Scale
- The total reading on the scale will be the sum of the weight of the pebbles and the rate of change of momentum:
\[ F = W + \Delta P \]
\[ F = (n \cdot t \cdot m \cdot g) + (n \cdot m \cdot \sqrt{2gh}) \]
- This simplifies to:
\[ F = n \cdot m \cdot (t \cdot g + \sqrt{2gh}) \]

### Final Answer
The scale reading at time \( t \) after the pebbles begin to fill the box is:
\[ F = n \cdot m \cdot (t \cdot g + \sqrt{2gh}) \]