Question

Estimate the uncertainty for measuring the coefficient of drag of 0. 1 on an object with a planform area A = 0. 5 m^2 as a function of velocity for velocities ranging from 1 m/sec to 100 m/sec (C_D = D/1/2 rho V^2 A) using a force balance that has a resolution of 1 N and a range of 1000N. The area is known with an uncertainty of 0. 15%, and the velocity is known with an uncertainty of 0. 1 m/s. The fluid density is inferred from the ideal gas law and where the temperature is known with an uncertainty of 1 degree C and the pressure is known with a certainty of 0. 2 kPa. Assume room temperature is 20 degree C and the pressure is atmospheric pressure

Answer 1

To estimate the uncertainty in measuring the coefficient of drag (C_D), we need to consider the uncertainties in the various parameters involved, including the force measurement, planform area, velocity, fluid density, and their respective relationships in the equation for C_D.

1. Uncertainty in force measurement:

The force balance used for the measurement has a resolution of 1 N and a range of 1000 N. The uncertainty in force measurement can be estimated as:

δF = ±(1/2) * (F_range / resolution) = ±(1/2) * (1000 / 1) = ±500 N

2. Uncertainty in planform area:

The planform area A is known with an uncertainty of 0.15%. Therefore, the uncertainty in A can be estimated as:

δA = ±(0.15/100) * A = ±0.00075 m^2

3. Uncertainty in velocity:

The velocity is known with an uncertainty of 0.1 m/s. Therefore, the uncertainty in velocity can be estimated as:

δV = ±0.1 m/s

4. Uncertainty in fluid density:

The fluid density can be inferred from the ideal gas law, assuming room temperature of 20°C and atmospheric pressure. The uncertainty in fluid density can be estimated using the following formula:

δρ = (δP/P + δT/T) * ρ

where δP is the uncertainty in pressure, δT is the uncertainty in temperature, and ρ is the fluid density. Assuming a pressure uncertainty of 0.2 kPa and a temperature uncertainty of 1°C, we get:

δρ = ((0.2/101.3) + (1/293)) * ρ = 0.0054 * ρ

5. Relationship between parameters:

Finally, we need to consider the relationship between the parameters in the equation for C_D. Using the formula for C_D = D/(1/2 ρ V^2 A), we can estimate the uncertainty in C_D as:

δC_D/C_D = √[(δD/D)^2 + (2δρ/ρ)^2 + (δV/V)^2 + (δA/A)^2]

where δD is the uncertainty in force measurement, δρ is the uncertainty in fluid density, δV is the uncertainty in velocity, and δA is the uncertainty in planform area.

Substituting the estimated uncertainties, we get:

δC_D/C_D = √[(500/1)^2 + (2*0.0054)^2 + (0.1/V)^2 + (0.00075/A)^2]

We can estimate the maximum uncertainty in C_D by substituting the maximum values for V and A:

δC_D/C_D = √[(500/1)^2 + (2*0.0054)^2 + (0.1/1)^2 + (0.00075/0.5)^2] = 1.10

Therefore, the estimated uncertainty in measuring the coefficient of drag is approximately ±10%.

Answer 2

To estimate the uncertainty for measuring the coefficient of drag (C_D) of an object with a planform area of A = 0.5 m² as a function of velocity, we need to consider the sources of uncertainty in the measurements of velocity, force, and area.

First, we need to calculate the range of expected drag force measurements. Using the given force balance with a resolution of 1 N and a range of 1000 N, the uncertainty in force measurements can be estimated to be ±0.5 N. For a given velocity, the drag force can be calculated using the formula: D = C_D * 0.5 * rho * V^2 * A, where rho is the fluid density, V is the velocity, and A is the planform area. The uncertainty in the planform area is given as 0.15%, which corresponds to ±0.00075 m². We can assume that the uncertainty in the fluid density is negligible compared to the other sources of uncertainty.

Next, we need to estimate the uncertainty in velocity measurements. The velocity is known with an uncertainty of 0.1 m/s, which corresponds to ±0.05 m/s. To estimate the range of expected drag force measurements, we can use the maximum and minimum values of the velocity range (1 m/s to 100 m/s) and the maximum and minimum values of the planform area uncertainty. This gives us a range of expected drag forces from ±0.026 N to ±526 N.

Finally, we can estimate the uncertainty in the coefficient of drag by dividing the uncertainty in drag force by the maximum possible drag force, which occurs at the highest velocity and with the maximum planform area uncertainty. This gives us an uncertainty in drag force of ±0.526 N. Dividing this by the maximum drag force of 1000 N gives us an uncertainty in the coefficient of drag of approximately ±0.00053.

Therefore, the uncertainty in the coefficient of drag for an object with a planform area of 0.5 m² as a function of velocity, measured using a force balance with a resolution of 1 N and a range of 1000 N, is approximately ±0.00053.