Question

Three particles $A,B,C$ 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 at point $A$ so that it can move without friction about the vertical axis through $A$. The body is set into rotational motion on the table about this axis 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 side $BC$ is parallel to x-axis, force $F$ is applied on $B$ along $BC$ (as in the figure). Obtain the x-component and the y-component of the force exerted by the hinge on the body, immediately after time $T$.

Answer 1

The centre of mass of the system is at the centroid of a triangular assembly. The CM moves along a circular path with constant angular velocity. Therefore, there must be a horizontal centripetal force directed towards the axis at the hinge

From figure we find

$AD=l\sin {60}^o=\frac{l\sqrt{3}}{2}$

$AO=\frac{2}{3}AD=\frac{l^2}{\sqrt{3}}=r$

the centripetal acceleration $a_c={\omega }^{2}r={\omega }^2\left(\frac{l}{\sqrt{3}}\right)$

Tangential acceleration $a_t=\alpha r=\alpha \left(\frac{l}{\sqrt{3}}\right)$

(b) Let $F_x$ and $F_y$ be the forces applied by the hinges along x-axis and y-axis respectively. The system is in non-centroidal rotation. The three equations of motion are

$\sum F_x=F_x+F=\left(3m\right)a_t=3m\left(\frac{l}{\sqrt{3}\alpha }\right)...\left(i\right)$

$\sum F_x=F_y+3m\left(\frac{l}{\sqrt{3}}\right){\omega }^2....\left(ii\right)$

$\sum \tau =F\times \left(\frac{\sqrt{3}}{2}l\right)=2m{l}^2\alpha ....\left(iii\right)$

From Eq (iii) $\alpha =\frac{\sqrt{3}F}{4ml}$

From Eq (i), $F_x+F=3m\left(\frac{l}{\sqrt{3}\alpha }\right)\times \frac{\sqrt{3}F}{4ml}=\frac{3F}{4}$

$\Rightarrow F_x=\frac{F}{4}$

from Eq. (ii) $F_y=\sqrt{3}ml{\omega }^2$