The spring of the pressure gauge shown in the figure below has a force constant of 1,080 N/m, and the piston has a radius of 1.43 cm. As the gauge is lowered into water, what change in depth causes the piston to move in by 0.750 cm?
k = 1080 N/m, r = 0.0143 m,
A = πr²= π•0.0143²=6.42•10⁻⁴ m²
The pressure P = •h,
where water density is ρ = 1000 kg/m³
g = 9.8 m/s² , x = 0.0075 m
For the elastic force of the spring
F = kx.
h = kx / A•ρ•g = …
To determine the change in depth that causes the piston to move in by 0.750 cm, we can use the concept of pressure in fluids and Hooke's law.
Here are the steps to calculate the change in depth:
1. Identify the relevant equations:
- Hooke's law: F = -kx, where F is the force, k is the force constant (spring constant), and x is the displacement.
- Pressure in a fluid: P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height (depth) of the fluid column.
2. Determine the force exerted by the spring:
According to Hooke's law, the force exerted by the spring is given by F = kx. Here, the force constant (k) is 1,080 N/m, and the displacement (x) is given as 0.750 cm. However, we need to convert x into meters before calculating the force.
x = 0.750 cm = 0.750 / 100 = 0.0075 m.
Thus, the force exerted by the spring is F = 1,080 N/m * 0.0075 m = 8.1 N.
3. Determine the pressure exerted on the piston:
The pressure exerted on the piston is equal to the force divided by the area of the piston. The area of the piston can be calculated using the formula for the area of a circle: A = πr^2, where r is the radius of the piston.
The given radius of the piston is 1.43 cm, which needs to be converted to meters.
r = 1.43 cm = 1.43 / 100 = 0.0143 m.
The area of the piston is A = π * (0.0143 m)^2 = 6.44 × 10^-4 m^2.
Therefore, the pressure exerted on the piston is P = F / A = 8.1 N / 6.44 × 10^-4 m^2.
4. Determine the density of water:
Water has a known density of ρ = 1000 kg/m^3.
5. Calculate the change in depth:
Using the equation for pressure in a fluid (P = ρgh), we can rewrite it as h = P / (ρg).
By substituting the values, we get h = (8.1 N / 6.44 × 10^-4 m^2) / (1000 kg/m^3 * 9.8 m/s^2).
Evaluating this expression, we find h = 1.32 m.
Therefore, the change in depth that causes the piston to move in by 0.750 cm is approximately 1.32 meters.