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Chapter 9: Problem 62

As is discussed in Section \(9.3,\) the drag on a rough golf ball may be less than that on an equal-sized smooth ball. Does it follow that a 10 -m-diameter spherical water tank resting on a \(20-\mathrm{m}\) -tall support should have a rough surface so as to reduce the moment needed at the base of the support when a wind blows? Explain.

Short Answer

Expert verified
No, the presence of a rough surface on the spherical water tank does not necessarily lead to a reduction in the moment needed at the base of the support when wind blows, despite the fact a rough surface has less drag on a smaller, fast-moving object like a golf ball. The two scenarios have different dynamics - the water tank being stationary and large, while a golf ball is smaller and moving.

Step by step solution

01

Understand Drag and Surface Texture

Recall that drag force depends on shape, surface area, and speed relative to the fluid, as well as fluid properties like density and viscosity. In the case of a golf ball, the drag decreases with a rough surface because this causes air to 'cling' more to the ball and decreases the turbulent wake, and hence the drag. This is due to the ball's speed and the manner in which it cuts through the air.
02

Consider the Sphere

A 10-m-diameter spherical water tank resting on a 20-m-tall support isn't in motion, but wind blowing across it creates relative motion. However, unlike a golf ball, this tank is stationary and the wind blows past it, creating air resistance.
03

Delving into Torque

Torque is the rotational equivalent of linear force. The direction of the torque is determined by the axis of rotation and the direction of force. The moment needed at the base of the support to resist the wind relies on the torque created by the wind.
04

Compare the Scenarios

While a rough surface of a golf ball can reduce drag in its specific situation, the rough surface on the spherical water tank does not mean that it will have less drag, and hence torque. This is primarily because the two scenarios are quite different in their dynamical properties.

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Most popular questions from this chapter

An automobile engine has a maximum power output of \(70 \mathrm{hp},\) which occurs at an engine speed of \(2200 \mathrm{rpm} . \mathrm{A} 10 \%\) power loss occurs through the transmission and differential. The rear wheels have a radius of 15.0 in. and the automobile is to have a maximum speed of 75 mph along a level road. At this speed, the power absorbed by the tires because of their continuous deformation is 27 hp. Find the maximum permissible drag coefficient for the automobile at this speed. The car's frontal area is \(24.0 \mathrm{ft}^{2}\)

A laminar boundary layer velocity profile is approximated by \(u / U=2(y / \delta)-2(y / \delta)^{3}+(y / \delta)^{4}\) for \(y \leq \delta,\) and \(u=U\) for \(y >\delta .\) (a) Show that this profile satisfies the appropriate boundary conditions. (b) Use the momentum integral equation to determine the boundary layer thickness, \(\delta=\delta(x)\). Compare the result with the exact Blasius solution.

A full-sized automobile has a frontal area of \(24 \mathrm{ft}^{2},\) and a compact car has a frontal area of \(13 \mathrm{ft}^{2}\). Both have a drag coefficient of 0.5 based on the frontal area. Find the horsepower required to move each automobile along a level road in still air at 55 mph. Assume that the power required to deform the tires continuously at this speed (called rolling resistance) is equal to the power to overcome the air resistance. Estimate the gas mileage of both automobiles if the energy supplied to the drive wheels is \(\frac{1}{4}\) that available in the fuel. A gallon of fuel has \(1.0 \times 10^{8} \mathrm{ft} \cdot 1 \mathrm{b}\) available energy.

Typical values of the Reynolds number for various animals moving through air or water are listed below, For which cases is inertia of the fluid important? For which cases do viscous effects dominate? For which cases would the flow be laminar; turbulent? Explain. . $$\begin{array}{lcc} \text { Animal } & \text { Speed } & \text { Re } \\ \hline \text { (a) large whale } & 10 \mathrm{m} / \mathrm{s} & 300,000,000 \\\ \text { (b) flying duck } & 20 \mathrm{m} / \mathrm{s} & 300,000 \\ \text { (c) large dragonfly } & 7 \mathrm{m} / \mathrm{s} & 30,000 \\ \text { (d) invertebrate larva } & 1 \mathrm{mm} / \mathrm{s} & 0.3 \\ \text { (e) bacterium } & 0.01 \mathrm{mm} / \mathrm{s} & 0.00003 \end{array}$$

A \(1.2-1 b\) kite with an area of \(6 \mathrm{ft}^{2}\) flies in a \(20-\mathrm{ft} / \mathrm{s}\) wind such that the weightless string makes an angle of \(55^{\circ}\) relative to the horizontal. If the pull on the string is 1.5 lb, determine the lift and drag coefficients based on the kite area.
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