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Biomechanics 427 2023 Problem Set 6

Values and constants

Density of air, ρair = 1 kg m-3

Density of water, ρwater = 1000 kg m-3

Gravitational acceleration, g = 10 m s-2

π = 3.14

Human dimensions

    mass, m = 70 kg

    total body height, Htotal= 2 m

    body diameter, Dbody = 0.3 m

    chest height, Hchest = 1.5 m

    waist height, Hwaist = 1.0 m

    drag coefficient, Cd= 1.0

Equations of interest

Drag = 1 2 ρU 2ACd for high Reynolds number, A is the planform area Bernoulli’s Principle: p + ρgh + ½ρu2 = constant along a streamline

1. (3 pts) When making new burrows on the western plain, prairie dogs get the hole family to pitch in! Prairie dogs construct elaborate burrow systems that require ventilation. The burrows typically have an elevated mound entrance and a second one that is either level with the ground or even slightly depressed. Wind blowing across the prairie is deflected upward around the elevated mounds. For the questions below, you can assume the burrow is not obstructed by an animal (as it is in the figure to the right). You can also assume the freestream velocity of the undisturbed wind (far from the mound) is 3 m/s and at the very top of the mound reaches, it reaches 5 m/s.

(a) Explain in words why the wind speed is greater at the top of the mound relative to the ground upstream of the mound.

(b) Using Bernoulli’s principle, and ignoring the changes in potential energy changes of the wind, what is the pressure difference between the two burrow openings in the figure above?

(c) Does the flow exit through the mound or through the lower portion, why?

2. (3 pts) Port or starboard? Either oar. A rower pulls back on an oar to propel their boat through water. In this problem, consider the oar to be a paddle (circular disc) with an area of 0.2 m2 and a drag coefficient (Cd) of 1.0. It takes one second for the oar to complete a stroke (a swing through a 90-degree arc centered about the mid-section of the boat) with a constant angular velocity. The distance from the point of rotation of the paddle (to the center of the disc) is 2 m.

(a) Derive an equation that predicts how the thrust (forward force) in a stroke changes as a function of time if the boat’s forward velocity is assumed to be tiny with respect to the rearward oar velocity (i.e. the forward velocity of the boat is negligible).

(b) Plot the thrust as a function of time (0 to 1 sec)

(c) How might the fact that the boat is moving in a direction opposite to the oar modify your analysis? (hint: consider the velocity of the oar relative to the water).

3. (3 pts) If they add a clock to the leaning tower of Pisa, they’ll have the time and the inclination. One well-known impact of climate change is the increase in the frequency and severity of extreme weather events. Indeed, Seattle has seen some fairly dramatic winds over the past years, typically from extratropical cyclones that form in the Pacific Basin. Just standing in these gale force winds is challenging and, to avoid toppling over, one leans into the wind. For the purposes of this problem, assume the drag on a leaning human is centered at the chest height and the center of mass is located at the waist height. Moreover, assume the human is approximately cylindrical in shape. Use the appropriate equation for drag indicated above in addressing the questions that follow.

(a) Draw a diagram of forces and moments for a person leaning into the wind at an angle θ.

(b) Derive an equation that relates the angle (θ) to which a person must tilt to be perfectly balanced by a given wind speed (e.g., the moment generated by drag is to equal the moment generated by the person’s weight. Note: (i) drag is a function of wind speed (ii) the projected area of a vertical human is approximately rectangular (height x diameter) and (iii) the height used to estimate projected area is a function of leaning angle. Your equation should express wind speed U as a function of tilt angle θ and include any of the values/constants/dimensions shown at the top of page 1. Do not plug in values quite yet!

(c) Next, plug in values/constants/dimensions to generate an equation for wind speed U as a function of tilt angle θ.

(d) Plot the tilt angle as a function of wind speed for winds up to 25 m/s (~56 mph)

4. 1 pt. Go ahead and mako my day. In the paper by Domel et al. 2018 (see pdf in Canvas), the authors used shark skin as inspiration for the design of aerofoils, such as those used in blades of wind turbines, helicopters and drones. The authors argue that a denticle that is inspired by, rather than reproduces exactly, the geometry of the shark’s denticle is preferrable for many aquatic and aerospace applications. What is their rationale?