A 8.52 -kg block of metal is suspended from a scale and immersed in water as in Figure P9.30. The dimensions of the block are 12.0 cm x 10.0 cm x 10.0 cm. The 12.0-cm dimension is vertical, and the top of the block is 9.0 cm below the surface of the water. (a) What are the forces exerted by the water on the top and bottom of the block? (Take P0 = 1.013×105 N/m2.) (top force, bottom force)
To find the forces exerted by the water on the top and bottom of the block, we need to consider the pressure exerted by the water at those points.
The pressure exerted by a fluid at a given depth can be calculated using the formula:
P = P0 + ρgh
Where:
P is the pressure at a given depth
P0 is the atmospheric pressure at the surface (1.013×10^5 N/m^2)
ρ is the density of the fluid (water in this case)
g is the acceleration due to gravity (9.8 m/s^2)
h is the depth of the point below the surface
Let's calculate the forces exerted by the water on the top and bottom of the block.
First, let's find the pressure at the top of the block. The top of the block is located 9.0 cm below the surface of the water.
h_top = 9.0 cm = 0.09 m (converting cm to m)
Using the formula, we can calculate the pressure at the top:
P_top = P0 + ρgh_top
Next, let's find the pressure at the bottom of the block. The depth from the surface to the bottom of the block can be found by subtracting the height of the block from the depth of the top:
depth_bottom = depth_top + height_block
depth_bottom = 0.09 m + 0.10 m = 0.19 m
Using the formula, we can calculate the pressure at the bottom:
P_bottom = P0 + ρgh_bottom
Now, let's calculate the forces exerted by the water on the top and bottom of the block.
The force can be calculated using the formula:
F = P × A
Where:
F is the force exerted by the water
P is the pressure exerted by the water
A is the area over which the force is exerted
The area over which the force is exerted is the area of the top and bottom faces of the block, which can be calculated by multiplying the length and width of the block.
A_top = length_block × width_block
A_bottom = length_block × width_block
Substituting the values into the formula, we can find the forces:
F_top = P_top × A_top
F_bottom = P_bottom × A_bottom
So the forces exerted by the water on the top and bottom of the block can be calculated using the formulas above.
To find the forces exerted by the water on the top and bottom of the block, we need to consider the pressure exerted by the water at those points.
Step 1: Find the pressure exerted by the water:
The pressure exerted by a fluid at a certain depth is given by the formula:
P = P0 + ρgh
where:
P is the absolute pressure (Pa),
P0 is the atmospheric pressure (Pa),
ρ is the density of the fluid (kg/m3),
g is the acceleration due to gravity (m/s2), and
h is the depth of the point below the surface of the fluid (m).
Given:
P0 = 1.013 × 105 N/m2 (atmospheric pressure)
ρ = density of water = 1000 kg/m3 (density of water)
g = 9.8 m/s2 (acceleration due to gravity)
h = 9.0 cm = 0.09 m (depth of the top of the block)
Substituting the values into the formula, we can find the pressure at the top of the block:
P(top) = P0 + ρgh
P(top) = 1.013 × 105 N/m2 + (1000 kg/m3) × (9.8 m/s2) × (0.09 m)
P(top) = 1.013 × 105 N/m2 + 882 N/m2
P(top) ≈ 1.021 × 105 N/m2
Step 2: Calculate the force exerted by the water on the top and bottom of the block:
The force exerted by the water is given by multiplying the pressure by the area. Since the area of the top and bottom of the block is the same, we can assume that the force exerted on both sides will be equal.
Given:
Area = 10.0 cm × 10.0 cm = (10.0 cm)2 = (0.1 m)2 = 0.01 m2 (area of top/bottom)
Now we can calculate the forces:
Force(top) = P(top) × Area
Force(top) = (1.021 × 105 N/m2) × 0.01 m2
Force(top) = 1021 N
Force(bottom) = P(bottom) × Area (assuming P(bottom) ≈ P(top))
Force(bottom) ≈ Force(top)
Force(bottom) ≈ 1021 N
Therefore, the forces exerted by the water on the top and bottom of the block are approximately 1021 N each.