The spring of the pressure gauge has a force constant of 1,000 N/m,and the piston has a diameter of 2.00 cm.when the gauge is lowered into water,at what depth does the piston move in by 0.500 cm?
If the piston moves 0.500 cm = 5*10^-3 m, , a force of 5 N is applied to the piston.
The piston area is (pi/4)*(0.02)^2 = 3.14*10^-4 m^2
The pressure is then
5 N/(Piston Area, m^2) = ___ Pascals
Finally calculate the depth necessary to attain that gauge pressure.
To determine at what depth the piston moves in when the pressure gauge is lowered into water, we need to consider the force exerted by the water on the piston.
We can start by calculating the force exerted by the spring on the piston. The force exerted by the spring can be determined using Hooke's Law:
F = k * x
Where:
F is the force exerted by the spring
k is the force constant of the spring (k = 1,000 N/m)
x is the displacement of the piston from its initial position (x = 0.500 cm = 0.005 m)
Substituting the values into the equation, we get:
F = (1,000 N/m) * (0.005 m)
F = 5 N
The force exerted by the water on the piston can be calculated using the formula:
F = P * A
Where:
F is the force exerted by the water on the piston
P is the pressure exerted by the water
A is the area of the piston
We know the diameter of the piston is 2.00 cm, which means the radius (r) is 1.00 cm = 0.01 m. Thus, the area of the piston can be calculated as:
A = π * r^2
A = π * (0.01 m)^2
A = 0.000314 m^2
Substituting the values into the equation, we have:
5 N = P * 0.000314 m^2
To find P, we can rearrange the equation:
P = 5 N / (0.000314 m^2)
P ≈ 15,924 Pa
Since pressure increases with depth in a fluid, we can use the equation for hydrostatic pressure to find the depth (h) corresponding to this pressure:
P = ρ * g * h
Where:
P is the pressure exerted by the fluid (15,924 Pa)
ρ is the density of the fluid (water) ≈ 1,000 kg/m^3
g is the acceleration due to gravity (9.8 m/s^2)
h is the depth in meters
Substituting the values into the equation, we can solve for h:
15,924 = 1,000 * 9.8 * h
h = 15,924 / (1,000 * 9.8)
h ≈ 1.62 m
Therefore, when the gauge is lowered into water, the piston moves in by approximately 0.500 cm at a depth of 1.62 meters.