Example $1$

A rotating shaft with two weights can be modelled as shown in the Figure below. Weight $W_1=10N$@$30$ at a radius of $15$ cm and weight $W_2=15N$ @ $270$ at a radius of $10$ cm $.$ Assume the acceleration due to gravity is $g=10\frac{m}{s^2}.$ If only static balancing is done using a weight of $W_b=10N,$ find the angle ${}_b$ of the balance mass and its radius $R_b.$

Example $2$

Dynamic balancing is to be performed on the same rotating shaft as shown in the figure. For this case it is necessary that

a$)$ the shaft has constant angular acceleration and $1$ balancing mass in plane $4$

b$)$ the shaft has zero angular acceleration and $1$ balancing mass in plane $3$

c$)$ the shaft has zero angular acceleration and $2$ balancing masses in planes $3$ and $4$

d$)$ all of the above