Answer
(a) The rider must begin hanging on when $a_{max}$ equals g.
(b) $0.14m$
Work Step by Step
(a) We know that the downward force on the horse is $mg$. When the horse begins to accelerate downward at a rate greater than the value of $g$, then the rider must begin hanging to prevent separating from the horse.
(b) We can find the required amplitude as follows:
$\omega=\frac{2\pi}{T}=\frac{2\pi}{0.74s}=8.5rad/s$
Now $a_{max}=A\omega^2$
$\implies A=\frac{a_{max}}{\omega^2}$
We plug in the known values to obtain:
$A=\frac{9.81m/s^2}{(8.5rad/s)^2}$
$A=0.14m$
