Question 14.1: The Water Bed The mattress of a water bed is 2.00 m long by ...
The Water Bed
The mattress of a water bed is 2.00 m long by 2.00 m wide and 30.0 cm deep.
(A) Find the weight of the water in the mattress.
(B) Find the pressure exerted by the water bed on the floor when the bed rests in its normal position. Assume the entire lower surface of the bed makes contact with the floor.
(A) Conceptualize Think about carrying a jug of water and how heavy it is. Now imagine a sample of water the size of a water bed. We expect the weight to be relatively large.
Categorize This example is a substitution problem.
Find the volume of the water filling the mattress:
V=\ell w hUse Equation 1.1 and the density of fresh water (see Table 14.1) to find the weight of the water bed:
\rho \equiv \frac{m}{V} (1.1)
M g=(\rho V) g=\rho g \ell w hSubstitute numerical values:
\begin{aligned}M g & =\left(1 000 kg/m^3\right)\left(9.80 m/s^2\right)(2.00 m)(2.00 m)(0.300 m) \\& =1.18 \times 10^4 N\end{aligned}which is approximately 2 650 lb. (A regular bed, including mattress, box spring, and metal frame, weighs approximately 300 lb.) Because this load is so great, it is best to place a water bed in the basement or on a sturdy, well-supported floor.
(B) When the water bed is in its normal position, the area in contact with the floor is A=\ell w. Use Equation 14.1 to find the pressure:
P \equiv \frac{F}{A} (14.1)
P=\frac{M g}{\ell_w}=\frac{1.18 \times 10^4 N}{(2.00 m)(2.00 m)}=2.94 \times 10^3 Pa| Table 14.1 Densities of Some Common Substances at Standard Temperature (0°C) and Pressure (Atmospheric) |
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| Substance | ρ (kg/m³) | Substance | ρ (kg/m³) |
| Air | 1.29 | Iron | 7.86 × 10³ |
| Air (at 20°C and | Lead | 11.3 × 10³ | |
| atmospheric pressure) | 1.20 | Mercury | 13.6 × 10³ |
| Aluminum | 2.70 × 10³ | Nitrogen gas | 1.25 |
| Benzene | 0.879 × 10³ | Oak | 0.710 × 10³ |
| Brass | 8.4 × 10³ | Osmium | 22.6 × 10³ |
| Copper | 8.92 × 10³ | Oxygen gas | 1.43 |
| Ethyl alcohol | 0.806 × 10³ | Pine | 0.373× 10³ |
| Fresh water | 1.00 × 10³ | Platinum | 21.4 × 10³ |
| Glycerin | 1.26 × 10³ | Seawater | 1.03 × 10³ |
| Gold | 19.3 × 10³ | Silver | 10.5 × 10³ |
| Helium gas | 1.79 × 10^{-1} | Tin | 7.30 × 10³ |
| Hydrogen gas | 8.99 × 10^{-2} | Uranium | 19.1 × 10³ |
| Ice | 0.917 × 10³ | ||