EXSS 323 Week 9/10

EXSS 323 Week 9 & 10 Summary

Main Ideas of Weeks 9 and 10:

We have ignored frictional forces acting on objects because these forces are often relatively small and because understanding an "ideal" situation without friction is much easier. The real situation involving friction and other forces is somewhat more complicated.
Motion through media like air or water is affected by principles described by "fluid mechanics." These principles apply to motion through any medium which "flows" around an object.
Force resisting motion through a medium depends on surface and shape of an object.
Shape of an object affects what is called "form drag" (or "profile drag").
Profile drag force can be determined using the formula:
```
                   2
           C  A p V
            d
    F   =  ----------
     d          2
```
where Cd is "drag coefficient" a unitless quantity which depends on the shape of an object.

A is the frontal area (silhouette area) in square meters.

p is greek letter rho, and is the density of the medium in kg per cubic meter.

V is the flow velocity or relative velocity of the medium with respect to the object.
Example: What is the drag force acting on a sprinter moving at 10 m/s? Estimate values involved:
Cd = 0.8

A = 1 m² (approximately)
p = 1.2 kg/m³ (at sea level)
2 (.8)(1)(1.2)(10) drag force = ------------------ = 48 N 2

At a running speed of 5 m/s this runner will have a drag force considerably less than 48 N (plug in to the equation and see that the drag force would be 12 N). Notice that doubling the running speed from 5 to 10 m/s results in a four times increase in drag force (12 to 48 N).

Terminal Velocity: maximum speed that a person can attain when moving under the influence of gravity through a fluid like air or water. Terminal Velocity occurs when the gravitational force driving the motion is exactly balanced by drag force opposing the motion.

       F         =  F
        gravity      drag

                            2
                    C  A p V
                     d
m g sin(angle)   =  ----------
                        2

This expression can be rearranged to solve for V.

Example: a skier descending straight down a slope will accelerate under the influence of gravity. As the skier's velocity increases, the drag force will increase. This acceleration will continue until the drag force exactly balances the gravitational force. At that point the skier will be in equilibrium and will no longer accelerate.

If skier characteristics are:

mass = 70 kg

drag coefficent = 0.7

frontal area = 0.5 m²

density = 1 kg/m³

slope of 45 degrees

            2 m g sin(45)
V  =  sqrt[---------------] 
               C  A p
                d

           (2)(70)(9.8)(sin(45))
   =  sqrt[----------------------]
               (0.7)(0.5)(1)

   =  52.6 m/s

Note that this is a very good estimation of speed skiing world records which are greater than 200 km/hr (or about 56 m/s).

Bernoulli's Principle: flow velocity of a fluid over a surface is inversely proportional to the pressure exerted on that surface. Thus as velocity increases pressure decreases or as velocity decreases pressure increases.

Magnus Effect: a spinning object (like a baseball) will have slightly different flow velocities on each side due to the combination of forward motion and spinning. These different velocities result in differing pressures and a force causing curvature of path.

Examples of Magnus Effect: various baseball pitches. Fastball pitches tend to have "backspin" which results in an upward directed force. Curveball pitches have an obliquely angled "forward spin" resulting in a force causing additional curvature downward and to the side.

Lift force: also due to Bernoulli's principle. An airfoil shape (like a wing) will force air to flow more rapidly across the top than across the bottom. This results in higher pressure below than on top of the airfoil and an upward directed "lift" force.

Lift and Drag forces are also affected by the "angle of attack" of an object moving through a medium. The lift and drag forces are always perpendicular to each other but will have varying magnitudes as an object's angle of attack is changed.

The lift and drag forces are affected by the medium involved. Water has a much higher density than air (about 1000 vs 1 kg/m³) therefore drag and lift forces for a given speed would be much larger in water than in air.

Buoyant force in a fluid is an upward directed force which has a magnitude equal to the weight of the fluid displaced by an object. An object submerged will have gravity and buoyant force acting on it. If gravitational force acting downward is greater magnitude than buoyant force acting upward, the object will sink. If buoyant force is greater, the object will float.