A dump truck is being filled with sand. The sand falls straight downward from rest from a height of 2.00 m above the truck bed, and the mass of sand that hits the truck per second is 55.0 kg/s. The truck is parked on the platform of a weight scale. By how much does the scale reading exceed the weight of the truck and sand?

Options

A. 626 N
B. 539 N
C. 214 N
D. 344 N

Answer

To solve the problem, we need to determine how much the scale reading exceeds the weight of the truck and the sand. We will follow these steps:

### Step 1: Calculate the velocity of the sand just before it hits the truck.
The sand falls from a height of 2.00 m. We can use the formula for the velocity of an object falling freely under gravity:
\[ v = \sqrt{2gh} \]
where:
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( h = 2.00 \, \text{m} \)

Substituting the values:
\[ v = \sqrt{2 \times 9.8 \times 2.00} \]
\[ v = \sqrt{39.2} \]
\[ v \approx 6.26 \, \text{m/s} \]

### Step 2: Calculate the momentum of the sand hitting the truck per second.
The mass of sand hitting the truck per second is given as \( \dot{m} = 55.0 \, \text{kg/s} \). The momentum \( p \) of the sand hitting the truck is given by:
\[ p = \dot{m} \times v \]
Substituting the values:
\[ p = 55.0 \, \text{kg/s} \times 6.26 \, \text{m/s} \]
\[ p \approx 344.3 \, \text{kg m/s} \]

### Step 3: Calculate the force exerted by the sand on the truck.
The force exerted by the sand on the truck can be calculated using the change in momentum per unit time. Since the sand comes to a stop upon hitting the truck, the force \( F \) is equal to the rate of change of momentum:
\[ F = \frac{dp}{dt} = \dot{m} \times v \]
From Step 2, we have already calculated \( p \) as approximately 344.3 kg m/s. Therefore, the force is:
\[ F \approx 344.3 \, \text{N} \]

### Step 4: Calculate the total weight on the scale.
The total weight on the scale is the sum of the weight of the truck and the additional force due to the falling sand. If we denote the weight of the truck as \( W_t \) (which we do not know), the total force \( F_t \) on the scale is:
\[ F_t = W_t + F \]
Since we are interested in how much the scale reading exceeds the weight of the truck and sand, we can express this as:
\[ F_t - W_t = F \]

### Step 5: Conclusion
The scale reading exceeds the weight of the truck and sand by the force exerted by the sand:
\[ F_t - W_t = 344.3 \, \text{N} \]

Thus, the scale reading exceeds the weight of the truck and sand by approximately **344.3 N**.