Problem 81 - Physics

Question

A water bed has dimensions of \(1.83 \mathrm{~m} \times 2.13 \mathrm{~m} \times 0.229 \mathrm{~m}\). The floor of the bedroom will tolerate an additional weight of no more than \(6660 \mathrm{~N}\). Find the weight of the water in the bed and determine whether it should be purchased.

Step-by-Step Solution

Answer

The weight of the water bed is 8760.288 N, exceeding the 6660 N limit; it shouldn't be purchased.

Step 1: Calculate Volume of the Water Bed

To calculate the volume of the water bed, use the formula for the volume of a rectangular prism: \[ V = ext{length} \times ext{width} \times ext{height} \]Substitute the given dimensions:\[ V = 1.83 \times 2.13 \times 0.229 \text{ m}^3 \]Calculate to find the volume:\[ V = 0.892383 \, \text{m}^3 \]

Step 2: Calculate the Mass of the Water

Next, find the mass of the water using the formula:\[ ext{mass} = ext{density} \times ext{volume} \]The density of water is typically \( 1000 \, \text{kg/m}^3 \). Using this, calculate the mass:\[ ext{mass} = 1000 \, \text{kg/m}^3 \times 0.892383 \, \text{m}^3 = 892.383 \, \text{kg} \]

Step 3: Calculate the Weight of the Water

Now, calculate the weight of the water using the formula:\[ ext{weight} = ext{mass} \times g \]where \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity. Calculate the weight:\[ ext{weight} = 892.383 \, \text{kg} \times 9.81 \text{ m/s}^2 = 8760.288 \, \text{N} \]

Step 4: Determine Feasibility of Water Bed Purchase

Compare the weight of the water bed against the maximum additional weight the floor can tolerate:\[ 8760.288 \, \text{N} > 6660 \, \text{N} \]Since the weight of the water bed exceeds the maximum tolerance of the floor, the water bed should not be purchased. The floor cannot safely support the additional weight.

Key Concepts

Volume Calculation

Volume calculation plays a vital role in determining how much space an object occupies, and it directly impacts other physical properties like weight. When dealing with objects with rectangular shapes, such as a water bed, we use the formula for the volume of a rectangular prism.
The formula for calculating the volume is:

• Volume (V) = length × width × height

Given the dimensions of a water bed as 1.83 meters in length, 2.13 meters in width, and 0.229 meters in height, you can calculate the volume by multiplying these three measurements:
\[ V = 1.83 imes 2.13 imes 0.229 \text{ m}^3 \approx 0.892383 \, \text{m}^3 \]This calculated volume represents the total amount of space inside the water bed that can be filled with water, which is essential for finding out its weight.

Density of Water

Understanding the density of water is crucial when calculating the mass and weight of water in any given space, like a water bed. Density is defined as the mass per unit volume of a substance and is usually expressed in kilograms per cubic meter (\text{kg/m}^3). For water, this value is typically:

• Density of water = 1000 \text{ kg/m}^3

Using the density, you can calculate the mass of the water in the bed. Applying the formula:

• Mass = Density × Volume

Plugging in our numbers, we multiply the density of water by the previously calculated volume of the water bed:\[ \text{Mass} = 1000 \times 0.892383 = 892.383 \text{ kg} \]This mass represents how much water the bed is holding, which is necessary for determining the pressure or force it exerts on the floor.

Weight Calculation

Weight is the force exerted by gravity on an object, and it plays a crucial role in deciding if the water bed can be safely placed inside a room. To calculate weight, you multiply the object's mass by the acceleration due to gravity:

• Weight = Mass × Acceleration due to gravity (g)

In our case, after calculating the mass as 892.383 kg, we use the standard acceleration due to gravity, which is:

• g = 9.81 \text{ m/s}^2

Therefore, the calculation becomes:\[ \text{Weight} = 892.383 \times 9.81 = 8760.288 \text{ N} \]This value indicates the total force that the water inside the bed would exert on the floor, helping us compare it with the floor's weight tolerance.

Acceleration Due to Gravity

Acceleration due to gravity (\text{g}) is a constant that expresses the force exerted by Earth's gravitational pull on objects. On Earth, this value is approximately:

• g = 9.81 \text{ m/s}^2

This constant is crucial for converting mass into weight, as weight is directly affected by gravitational pull. When calculating the weight of an object from its mass, this factor ensures that the force is measured accurately.
Consider the mass of our water bed that we computed (892.383 kg). Multiplying it by the gravitational constant gives us:\[ \text{Weight} = 892.383 \times 9.81 = 8760.288 \text{ N} \]This calculation demonstrates how vital gravity is in determining the feasibility of placing a water bed in a room. It ensures that the computed weight aligns with physical laws, facilitating safe and informed decisions regarding the structural capacity of a building to support additional loads.