$1.$ In the set$-$up shown, the length of the string between the vibrator and the pulley is $1\mathrm{.}2$ m and the linear density is $1\mathrm{.}6$g$//$m$.$ The frequency of the string vibrator is set to $120$ Hz $.$

a$)$ How much mass does the weight have to have for the string to vibrate in the fundamental mode?

$2.$ The figure shows a person swinging on a rope, starting

at a point higher than A$.$

Consider the following distinct forces:

A downward force of gravity.

A force exerted by the rope pointing from A to O $.$

A force in the direction of the boy's motion.

A force pointing from O to A$.$

Which of the above forces is $($are$)$ acting on the boy

when he is at position A$?$ Briefly explain.