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challenge in mechanics 2

Two massive identical rods are joined by a light string attached to their midpoints & are also pivoted through a smooth pin passing through one end of each rod. The entire structure looks like the alphabet A. The system always remains in a vertical plane with the two remaining free ends of the rods on a smooth horizontal surface. The angle made by each rod with the horizontal is θ. If the string is suddenly cut find the initial acceleration of the pivot point.

Note by Ronak Pawar
3 years ago

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Challenge

Considering the left rod the acceleration of pivot point is downwards and the acceleration of the point of the rod which is in contact with the ground is leftwards. Hence the instantaneous centre of rotation can be located. Now $$\tau =\frac { mglcos\theta }{ 2 }=I\alpha$$. By using parellel axes theorom $$I=\frac { m{ l }^{ 2 } }{ 12 } +\frac { m{ l }^{ 2 } }{ 4 } =\frac { m{ l }^{ 2 } }{ 3 }$$
$$\Rightarrow \alpha =\frac { 3gcos\theta }{ 2l }$$

$$\Rightarrow { a }_{ Pivot }=\alpha \frac { l }{ 2 } \quad \boxed { \Rightarrow { a }_{ Pivot }=\frac { 3gcos\theta }{ 4 } }$$

Tell me whether this is correct or not. · 2 years, 12 months ago