The spring of the pressure gauge shown in the figure below has a force constant of 1,160 N/m, and the piston has a radius of 1.22 cm. As the gauge is lowered into water, what change in depth causes the piston to move in by 0.750 cm?
ρ(water) =1000 kg/m³ !!!!!!!!!!!!
(k•x)-(ρ•g•v) = zero
we have: k= 1160 N/m
x= .0075 m
ρ(water)= 1000 kg/m³
g= 9.8 m/s²
r= .0122 m
and v=h•A A= π•r² = π(.0122)²
h=??? (Change in the pressure)
so: h=(k•x)/(ρ•g•A)
=(1160*.0075)/(1000*9.8*(π(.0122)²)
h= 1.898557011 m :)
k = 1160 N/m
r = 1.22 cm = 0.0122 m
x = 0.750 cm =0.0075 m
Change in the pressure, when the gauge is taken into water
ΔP = F/ A= k•x /π•r² =
=1160•0.0075/ π•(0.0122)²= .....
ρ(water) =100 kg/m³
ΔP = ρwater• g •h,
h = ΔP/g• ρ(water)=…
(k•x)-(ρ•g•v) = zero
v=h•A
To determine the change in depth that causes the piston to move in by 0.750 cm, we need to apply the principles of fluid pressure and force.
Step 1: Determine the force applied by the spring:
The force applied by the spring can be calculated using Hooke's law:
F = k * x
where F is the force applied by the spring, k is the force constant, and x is the displacement of the piston.
Given:
Force constant (k) = 1,160 N/m
Displacement (x) = 0.750 cm = 0.0075 m
Substituting the given values into the formula:
F = 1,160 N/m * 0.0075 m
F = 8.7 N
Step 2: Determine the pressure difference between inside and outside of the piston:
The force applied by the spring is due to the pressure difference between the inside and outside of the piston. Using the formula:
ΔP = F / A
where ΔP is the pressure difference, F is the force applied, and A is the area of the piston.
Given:
Radius of the piston (r) = 1.22 cm = 0.0122 m
Substituting the given values into the formula:
A = π * r^2
A = π * 0.0122 m^2
ΔP = 8.7 N / (π * 0.0122 m^2)
ΔP ≈ 2,249.6694 Pa
Step 3: Determine the change in depth:
The pressure difference (ΔP) is caused by the change in depth. We can use the equation:
ΔP = ρ * g * h
where ΔP is the pressure difference, ρ is the density of water, g is the acceleration due to gravity, and h is the change in depth.
Given:
Acceleration due to gravity (g) = 9.8 m/s^2
Rearranging the equation to solve for h:
h = ΔP / (ρ * g)
The density of water (ρ) is approximately 1000 kg/m^3.
Substituting the given values into the equation:
h = 2,249.6694 Pa / (1000 kg/m^3 * 9.8 m/s^2)
h ≈ 0.2307 m
Therefore, the change in depth that causes the piston to move in by 0.750 cm is approximately 0.2307 meters.