A box is put on a scale that is marked in units of mass and adjusted to read zero when the box is empty. A stream of marbles is then poured into the box from a height h = 6.93 m above its bottom at a rate of R = 128 (marbles per second). Each marble has mass m = 7.14 g. If the collisions between the marbles and the box are completely inelastic, find the scale reading at time t = 5.90 s after the marbles begin to fill the box.

Question

A box is put on a scale that is marked in units of mass and adjusted to read zero when the box is empty. A stream of marbles is then poured into the box from a height h = 6.93 m above its bottom at a rate of R = 128 (marbles per second). Each marble has mass m = 7.14 g. If the collisions between the marbles and the box are completely inelastic, find the scale reading at time t = 5.90 s after the marbles begin to fill the box.

Answer 1

To find the scale reading at time t = 5.90 s, we need to consider the change in momentum and the change in the mass of the box over that time interval.

Let's break down the problem step by step:

1. Determine the total mass of the marbles poured into the box:
Total mass of marbles = mass of one marble * rate of pouring * time interval
= (7.14 g) * (128 marbles/s) * (5.90 s)

Note: Convert the mass of one marble to kg for consistency in units.

2. Determine the change in momentum of the marbles:
Change in momentum = total mass of marbles * change in velocity

In this case, the change in velocity is 6.93 m/s since the marbles fall freely from a height of 6.93 m.

3. Determine the change in momentum of the box:
The change in momentum of the box is equal in magnitude but opposite in direction to that of the marbles.

4. Calculate the scale reading:
The scale reading corresponds to the total momentum transferred to the box.

Scale reading = initial scale reading + change in momentum of the box.

Let's plug in the given values and calculate the scale reading:

1. Total mass of marbles:
Total mass of marbles = (7.14 g) * (128 /s) * (5.90 s) * (1 kg / 1000 g)

2. Change in momentum of the marbles:
Change in momentum of the marbles = Total mass of marbles * change in velocity

3. Change in momentum of the box:
Change in momentum of the box = - Change in momentum of the marbles

4. Calculate the scale reading:
Scale reading = initial scale reading + change in momentum of the box

By following these steps and performing the necessary calculations, you can find the scale reading at time t = 5.90 s after the marbles begin to fill the box.