Question 14.6: To make a car easier to find in crowded parking lots, a color...
To make a car easier to find in crowded parking lots, a colorful 2 in. diameter smooth plastic ball is attached to the end of the vehicle’s 3 ft antenna as shown in Figure 14.27. What is the bending moment on the antenna due to the ball if the car is moving at 50 mph?
We will calculate the drag force on the ball, then find the resulting bending moment. We will assume that the flow over the ball is the same as it would be without having the antenna nearby. Assuming air at 70°F,
Re=\frac{UD}{\nu}=\frac{(50\ mph)[1.47\ \mathrm{ft}/(mph-s)](2\ in.)(\mathrm{ft}/12 in.) }{1.64×10^{−4}\ \mathrm{ft}^2/s }=7.5\times 10^4
From Figure 14.24 we find CD = 0.5. The drag force on the ball is calculated next from FD = CD \frac{1}{2} ρU2A, where A = \piD2/4. Inserting the data, we find
F_D = (0.5)\left(\frac{1}{2} \right) (2.329×10^{−3}\ slug/\mathrm{ft}^3)(73.3\ \mathrm{ft}/s)^2 \frac{π(0.1667\ \mathrm{ft})^2}{4}
= 6.83 × 10−2 lbf
Ignoring any curvature of the antenna, the bending moment is
M = FDL = (6.83 × 10−2 lbf)(3 ft ) = 0.2 ft-lbf
Would you recommend adding roughness to the ball?