A 50-cm-diameter, 400 g beach ball is dropped with a 4.0 mg ant riding on the top. The ball experiences air resistance, but the ant does not. What is the magnitude of the normal force exerted on the ant when the ball’s speed is 2.0 m/s?

A 50-cm-diameter, 400 g beach ball is dropped with a 4.0 mg ant riding on the top. The ball experiences air resistance.

the ball’s speed is 2.0 m/s

diameter of the ball, D = 50 cm = 0.5m

radius of the ball, r = 0.25m

mass of the beach ball, m₁ = 400 g = 0.4 kg

mass of the ant, m₂ = 4 x 10⁻⁶ kg

speed of the ball, v = 2 m/s

The area of the ball, assuming spherical ball is given by;

A = 4πr²

A = 4π(0.25)² = 0.7854 m²

The drag force (resistance) experienced by the spherical ball is given as;

$F_D=\frac{1}{2}C\rho A{v}^2$

where;

C is the drag coefficient of the spherical ball = 0.45

ρ is density of air = 1.21 kg/m³

$\begin{array}{r}{F}_D=\frac{1}{2}C\rho A{v}^2\\ F_D=\frac{1}{2}\left(0.45\right)\left(1.21\right)\left(0.7854\right)(2{)}^2\\ F_D=0.855\mathrm{N}\end{array}$

The downward force of the ball due to its weight and that of the ant is given by;

role="math" localid="1650545692020" $\begin{array}{r}{F}_g=mg\\ F_g=g\left(m_{ant}+m_{ball}\right)\\ F_g=g\left(4\ast {10}^{-6}\mathrm{kg}+0.4\mathrm{kg}\right)\\ \left.F_g=g\left(0.400004\mathrm{kg}\right)\text{(mass of the ant is insignificant}\right)\\ F_g=9.8\left(0.4\right)\\ F_g=3.92\mathrm{N}\end{array}$

The net downward force experienced by the ball is given by;

$\begin{array}{r}{F}_{net}=F_g-F_D\\ F_{net}=3.92\mathrm{N}-0.855\mathrm{N}\\ F_{net}=3.07\mathrm{N}\end{array}$

This downward force experienced by the ball is equal to the normal reaction it exerts on the ant.

Thus, the normal force exerted on the ant is 3.07N.