A 1-m-tall barrel is filled with water (with a weight density of 9800 N/m3 ). Show that the water pressure on the bottom of the barrel is 9800 N/m2 or, equivalently, 9.8 kPa.
The correct answer and explanation is :
To calculate the water pressure on the bottom of the barrel, we can use the formula for hydrostatic pressure:
[
P = \rho g h
]
Where:
- ( P ) is the pressure at depth ( h ),
- ( \rho ) is the density of the fluid (in this case, water),
- ( g ) is the acceleration due to gravity,
- ( h ) is the depth of the fluid at the point where the pressure is being calculated (in this case, the depth is the height of the barrel).
Step-by-Step Calculation:
- Given Data:
- The height of the water in the barrel ( h = 1 \, \text{m} ),
- The weight density of water is ( \rho g = 9800 \, \text{N/m}^3 ),
- Where ( \rho g ) is the specific weight of water. The specific weight is equivalent to the weight per unit volume, which is a measure of weight density.
- The value of ( g ) (gravitational acceleration) is approximately ( 9.81 \, \text{m/s}^2 ).
- Substitute Values:
The pressure at the bottom of the barrel is directly given by the product of the weight density and the height of the water:
[
P = 9800 \, \text{N/m}^3 \times 1 \, \text{m} = 9800 \, \text{N/m}^2
]
So, the pressure on the bottom of the barrel is:
[
P = 9800 \, \text{N/m}^2 \, \text{or} \, 9.8 \, \text{kPa}
]
Explanation:
- Hydrostatic Pressure: In fluids, the pressure increases with depth due to the weight of the fluid above. The deeper you go, the greater the weight of the water, which exerts more pressure on the object at that depth.
- Pressure Units: The unit of pressure is the Newton per square meter (N/m²), commonly referred to as a Pascal (Pa). 1 kPa = 1000 Pa.
- In this case, the weight density of water is 9800 N/m³, and the height of the water column is 1 meter. Hence, the pressure exerted on the bottom of the barrel is 9800 N/m², which is equivalent to 9.8 kPa.
Here is the illustration showing the 1-meter tall barrel filled with water. The hydrostatic pressure on the bottom is clearly labeled as 9800 N/m², corresponding to the water pressure at that depth.