Problem 40 - College Physics

Question

The Magdeburg water bridge is a channel bridge over the River Elbe in Germany ( \(\mathbf{F i g .} 9.38\) ). Its dimensions are length \(918 \mathrm{~m}\), width \(43.0 \mathrm{~m},\) and depth \(4.25 \mathrm{~m}\). (a) When filled with water, what is the weight of the water? (b) What is the pressure on the bridge floor?

Step-by-Step Solution

Answer

(a) The weight of the water is 1658257500 N. (b) The pressure is 42000 Pa.

Step 1: Find the Volume of the Water

To find the weight of the water when the channel is full, we first need to calculate the volume of the water. The volume of a rectangular prism (like this channel) can be calculated using the formula: \[ \text{Volume} = \text{length} \times \text{width} \times \text{depth} \]. Substituting in the provided values gives us: \[ \text{Volume} = 918 \text{ m} \times 43.0 \text{ m} \times 4.25 \text{ m} \].

Step 2: Calculate the Volume

Calculate the volume using the values from Step 1: \[ 918 \times 43.0 \times 4.25 = 168597.5 \text{ cubic meters} \]. So, the volume of the water is 168597.5 cubic meters.

Step 3: Convert Volume to Weight

To convert the volume of water to weight, we use the fact that the density of water is approximately 1000 kg/m³. Thus, the mass of the water is \( 168597.5 \times 1000 \text{ kg} \). Since weight is mass times gravity (\( g \approx 9.8 \text{ m/s}^2 \)), the weight is \[ 168597500 \text{ kg} \times 9.8 \text{ m/s}^2 = 1658257500 \text{ N} \].

Step 4: Calculate the Pressure on the Bridge Floor

Pressure is defined as force per unit area. The pressure exerted by the water on the floor of the bridge is given by the equation: \[ P = \frac{F}{A} \], where \( F \) is the force (weight of the water) and \( A \) is the area. The area in this case is the width times the length of the bridge: \[ A = 918 \text{ m} \times 43.0 \text{ m} = 39474 \text{ m}^2 \].

Step 5: Compute the Pressure

Substitute the values into the pressure formula: \[ P = \frac{1658257500 \text{ N}}{39474 \text{ m}^2} \]. This simplifies to approximately \[ P = 42000 \text{ Pa} \].

Key Concepts

Volume Calculation

Calculating volume is essential in many physics problems, especially when dealing with objects that hold fluids like the Magdeburg water bridge. The first step is to identify the shape of the object. In this case, the bridge is a rectangular prism. To find the volume, we can use the formula:

• Volume \( = \text{length} \times \text{width} \times \text{depth} \)

All measurements need to be in meters to ensure the volume is in cubic meters (m³). For the water bridge, the length is 918 m, the width is 43.0 m, and the depth is 4.25 m. Plug these values into the formula:

• \( Volume = 918 \times 43.0 \times 4.25 = 168597.5 \text{ m}^3 \)

Thus, the volume of water that the bridge can hold is 168597.5 cubic meters.

Weight and Mass

To find the weight of something, knowing its mass is crucial. Mass is a measure of the amount of matter in an object, while weight is the force exerted by gravity on that mass. For water, mass is often derived from volume, given its density. The density of water is approximately 1000 kg/m³. Thus, the mass of the water can be calculated by multiplying its volume by this density:

• Mass \( = 168597.5 \times 1000 = 168597500 \text{ kg} \)

Weight is calculated by multiplying mass by the acceleration due to gravity \( (g \approx 9.8 \text{ m/s}^2) \):

• Weight \( = 168597500 \times 9.8 = 1658257500 \text{ N} \)

Here, \( N \) stands for Newton, the unit of force or weight.

Pressure Calculation

Pressure is the force applied per unit area. It's vital to understand how pressure functions, particularly in structures like a bridge carrying a weight of water. The formula for pressure \( P \) is:

• \( P = \frac{F}{A} \)

where \( F \) is the force (or weight) applied, and \( A \) is the area over which force is applied. For the Magdeburg water bridge, the area of the bridge’s floor can be calculated by multiplying its length and width:

• Area \( = 918 \times 43.0 = 39474 \text{ m}^2 \)

Substituting these values into the equation for pressure we get:

• \( P = \frac{1658257500}{39474} \approx 42000 \text{ Pa} \)

Pressure is hence approximately 42000 Pascals (Pa), showing how the force of water is distributed across the bridge's floor.

Density of Water

The density of a substance is its mass per unit volume. For water, a commonly used and significant value is 1000 kg/m³. This property helps in determining how much mass is present in a specific volume of water, which is essential in many calculations involving water either in physics problems or engineering projects.Understanding density allows us to predict whether objects will float or sink when placed in water and how much water weighs per unit volume. Given its importance, density is often a cornerstone in physics education touching upon topics ranging from buoyancy to hydrostatics. Applying this in our problem:

• Mass of water \( = \text{Volume} \times \text{Density} \)
• \( 168597.5 \text{ m}^3 \times 1000 \text{ kg/m}^3 = 168597500 \text{ kg} \)

This relationship provides a reliable foundation to use in various calculations, showing the versatility of understanding basic properties like the density of water.