The spring of the pressure gauge shown in the figure below has a force constant of 1 100 N/m, and the piston has a diameter of 1.00 cm. As the gauge is lowered into water in a lake, what change in depth causes the piston to move in by 0.550 cm?
What "figure below"?
Please describe it.
To determine the change in depth that causes the piston to move in, we need to consider the forces acting on the piston.
Here's how to approach this problem:
Step 1: Identify the forces acting on the piston.
In this case, there are two forces acting on the piston: the force exerted by the spring, and the force exerted by the water.
Step 2: Determine the force exerted by the spring.
The force exerted by the spring can be calculated using Hooke's Law, which states that the force applied by a spring is directly proportional to the displacement. The formula for Hooke's Law is given by:
F_spring = k * x
where F_spring is the force exerted by the spring, k is the force constant of the spring, and x is the displacement of the piston.
In this case, the force constant of the spring is given as 1,100 N/m, and the displacement of the piston is 0.550 cm (which needs to be converted to meters):
x = 0.550 cm = 0.550 * 10^(-2) m = 0.00550 m
Substituting these values into the equation, we have:
F_spring = 1,100 N/m * 0.00550 m
Step 3: Determine the force exerted by the water.
The force exerted by the water is equal to the pressure exerted by the water multiplied by the area of the piston.
The pressure exerted by the water can be calculated using the equation:
P = ρ * g * h
where P is the pressure, ρ is the density of water, g is the acceleration due to gravity, and h is the change in depth.
The area of the piston can be calculated using the formula for the area of a circle:
A = π * r^2
where A is the area, and r is the radius of the piston.
The diameter of the piston is given as 1.00 cm, which needs to be converted to meters:
d = 1.00 cm = 1.00 * 10^(-2) m = 0.01 m
The radius of the piston is half of the diameter:
r = 0.01 m / 2 = 0.005 m
Substituting these values into the equation, we have:
A = π * (0.005 m)^2
Step 4: Calculate the force exerted by the water.
The force exerted by the water can be calculated using the equation:
F_water = P * A
Substituting the values for pressure and area, we have:
F_water = (ρ * g * h) * (π * (0.005 m)^2)
Step 5: Set up the equilibrium equation.
Since the piston is not moving, the force exerted by the spring and the force exerted by the water must be equal. Therefore, we can set up the following equation:
F_spring = F_water
Step 6: Solve for the change in depth (h).
Rearrange the equation to solve for h:
h = (F_spring / (ρ * g * A))
Step 7: Substitute the given values and calculate the change in depth.
Substitute the given values into the equation, including the force exerted by the spring obtained in Step 2, the density of water (ρ = 1,000 kg/m³), the acceleration due to gravity (g = 9.8 m/s²), and the area of the piston obtained in Step 3:
h = (F_spring / (1,000 kg/m³ * 9.8 m/s² * (π * (0.005 m)^2)))
Finally, calculate the value of h to obtain the change in depth that causes the piston to move in by 0.550 cm.