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## Q. 14.6

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?

## Verified Solution

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 = $\pi$D2/4. 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?