A disc of mass 10 g is kept floating horizontally by throwing 10 marbles per second against it from below . If mass of each marble is 5 g Calculate the velocity with which marbles are striking the disc . Assume that marbles strike the disc . Normally and rebound downwards with the same speed.

Options

A. `2.98 ms^(-1)`
B. `0.98 ms^(-1)`
C. `0.49 ms^(-1)`
D. `1.96 ms^(-1)`

Explanation

To solve the problem, we need to calculate the velocity with which the marbles are striking the disc. Here’s a step-by-step breakdown of the solution:

### Step 1: Calculate the weight of the disc
The weight of the disc (W) can be calculated using the formula:
\[ W = m \cdot g \]
where:
- \( m = 10 \, \text{g} = 10 \times 10^{-3} \, \text{kg} \)
- \( g = 9.8 \, \text{m/s}^2 \)

Calculating this gives:
\[ W = 10 \times 10^{-3} \, \text{kg} \times 9.8 \, \text{m/s}^2 = 98 \times 10^{-3} \, \text{N} \]

### Step 2: Determine the change in momentum of the marbles per second
Each marble has a mass of \( 5 \, \text{g} = 5 \times 10^{-3} \, \text{kg} \). Since 10 marbles are thrown per second, the total mass of marbles per second (M) is:
\[ M = 10 \times 5 \times 10^{-3} \, \text{kg} = 50 \times 10^{-3} \, \text{kg} \]

When a marble strikes the disc with velocity \( V \) and rebounds with the same speed in the opposite direction, the change in momentum (Δp) for one marble is:
\[ \Delta p = \text{final momentum} - \text{initial momentum} = (-mV) - (mV) = -2mV \]
Thus, for 10 marbles per second:
\[ \Delta p_{\text{total}} = 10 \times (-2 \times 5 \times 10^{-3} \times V) = -100 \times 10^{-3} \times V \]

### Step 3: Relate change in momentum to force
The change in momentum per second is equal to the force exerted by the marbles on the disc. Therefore:
\[ F = 100 \times 10^{-3} \times V \]

### Step 4: Set the force equal to the weight of the disc
Since the force exerted by the marbles must balance the weight of the disc for it to float:
\[ 100 \times 10^{-3} \times V = 98 \times 10^{-3} \]

### Step 5: Solve for the velocity \( V \)
Rearranging the equation gives:
\[ V = \frac{98 \times 10^{-3}}{100 \times 10^{-3}} = 0.98 \, \text{m/s} \]

### Conclusion
The velocity with which the marbles are striking the disc is:
\[ V = 0.98 \, \text{m/s} \]