Question

A portable basketball set has a base and a post arrangement. The post arrangement
consists of a post, backboard, hoop and net. The base can be filled with water to
increase stability. The maximum weight of the base is about 810 N and the weight
of the post arrangement is 26.0 N. The basketball set may topple over when the
wind blows due to the large area of the backboard.
a) Determine the minimum force of the wind, Fw that will cause the basketball set
to be blown over when it is at the angle shown. Ignore the effect of the wind on
the base.
b) The base is filled with sand instead of water. The density of the sand is greater
than the density of the water. Justify what would happen to the value of Fw
calculated in part (a).

Answer 1

a) The torque the wind creates is

$$\tau_w=F_wh.$$

h=d+l - height between the base and the basket.

The torque that creates the base is

$$\tau_b=W_br\cos\theta.$$

r - the half of length of the base, W - weight of the base.

As the basketball set inclines just a little bit, the weight of the post arrangement starts creating its torque that helps the wind:

$$\tau_p=W_al\sin\theta.$$

As we see, the set topples over when the wind creates enough torque to rotate the base with the post for angle θ at which the torques of the post and base become equal:

$$W_br\cos\theta=W_al\sin\theta,\\\space\\ \theta=\arctan\frac{W_br}{W_al}=75°.$$

Until this angle, the magnitude of wind force required to topple over the set is decreasing.

So, the initial force of wind it

$$\tau_w=F_wh=\tau_b=W_br\cos0°,\\\space\\ F_w=\frac{W_br}{h}=\frac{W_br}{d+l}=86.2\text{ N}.$$

b) Since the density of sand is greater than density of water, W_b will increase and, according to the last equation, more force will be required.