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Calculate the weight of the ball of a diameter 80mm which is just supported in vertical air-stream, which is flowing at a velocity of 7m/s

Calculate the weight of the ball of a diameter 80mm which is just supported in vertical air-stream, which is flowing at a velocity of 7m/s. The density of the air is $1.25 kg/ m^3. \theta = 1.5 \times 10-4 m^2/sec.$

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Solution :

Given :

Diameter, $D=80 {~mm}=0.08 {~m}$

Velocity of air, $u=7{~m} / {s}$

Density of air, $\rho=1.25 {~kg} /{m}^{3}$

Kinematic viscosity, $\nu=1.5$ stokes $=1.5 \times 10^{-4} {~m}^{2}/{s}$

Reynold number (Re)

\begin {aligned}\quad R_{e} &=\frac{ \rho u D}{ \mu }=\frac{uD}{\nu}\\ &=\frac{7 \times .08}{1.5 \times 10^{-4}}\\ &=3730 \end {aligned}

$1000\lt R_e \lt10000$

Thus the value of $C_{D}=0.5$

Drag force $F_D$,

\begin {aligned} \quad F_{D}=C_{D} \times A \times \frac{\rho u^{2}}{2}\end{aligned} ....(i)

where A is area of ball

\begin{aligned} \quad A &=\frac{\pi}{4} D^{2}=\frac{\pi}{4}(0.08)^{2} \\ &=0.005026 {~m}^{2}\end{aligned}

From equation (i)

\begin{aligned} \quad F_{D} &=0.5 \times 0.005026 \times \frac{1.25 \times 7^{2}}{2}\\ &=0.07696 {~N} \end{aligned}

Weight of ball $F_{D}=0.07696 {~N}$.