Question

Three particles A , B and C, each of mass m, are connected to each other by three massless rigid rods to form a rigid , equilateral triangular body of side l. This body is placed on a horizontal frictionless table (x-y plane) and is hinged to it at the point A, so that it can move without friction about the vertical axis through A (see figure). The body is set into rotational motion on the table about/! with a constant angular velocity $\omega$.
(a) Find the magnitude of the horizontal force exerted by the hinge on the body.
(b) At time T, when the side BC is parallel to the x-axis, a force F is applied on B along BC (as shown). Obtain the x-component and the _y-component of the force exerted by the hinge on the body, immediately after time T.

Answer 1

(a) The distance of centre of mass (CM) of the system about point A will be
$r=\frac{l}{\sqrt{3}}$
Therefore, the magnitude of horizontal force exerted by the hinge on the body is
F = centripetal force or $F=(3m)r{\omega }^2$
or $F=(3m)\left(\frac{l}{\sqrt{3}}\right){\omega }^2$ or $F=\sqrt{3}ml{\omega }^2$
(b) Angular acceleration of system about point A is
$\alpha =\frac{{\tau }_A}{I_A}=\frac{(F)\left(\frac{\sqrt{3}}{2}l\right)}{2m{l}^2}=\frac{\sqrt{3}F}{4ml}$
Now, acceleration of CM along x-axis is
$a_x=r\alpha =\left(\frac{1}{\sqrt{3}}\right)\left(\frac{\sqrt{3}F}{4ml}\right)$ or $a_x=\frac{F}{4m}$
Let $F_x$ be the force applied by the hinge along x-axis.
Then, $F_x+F=(3m)a_x$
or $F_x+F=(3m)\left(\frac{F}{4m}\right)$
or $F_x+F=\frac{3}{4}F$
Further if $F_y$ be the force applied by the hinge along y-axis. Then,
$F_y$ = centripetal force
or $F_y=\sqrt{3}ml{\omega }^2$