Question 763: Centre of Mass and Rotational Motion

Question

A rigid uniform bar AB of length L is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $	heta$. Which of the following statements about its motion is/are correct?

Accepted Answer

We discuss the options as follows:
$\bullet$ For option (A) : When the bar makes an angle $	heta$ with the floor, the height of the midpoint is ${L \over 2}\cos 	heta $. this is height of centre of mass.
A force mg acts vertically downward. Thus, the midpoint of bar falls vertically downward. Hence, option (A) is correct.
$\bullet$ For option (B) : Here, we have $x = {L \over 2}\sin 	heta $ and $y = L\cos 	heta $; therefore,
$\sin 	heta  = {x \over {L/2}}$ and $\cos 	heta  = {y \over L}$
Using the condition ${\sin ^2}	heta  + {\cos ^2}	heta  = 1$, we get
${{{x^2}} \over {{{(L/2)}^2}}} + {{{y^2}} \over {{L^2}}} = 1$ (equation of ellipse)
Thus, trajectory of point A is not parabola. Hence, option (B) is incorrect.
$\bullet$ For option (C) : The torque acting about the point of contact with floor, that is, at point B is
$	au  = \overrightarrow F  	imes \overrightarrow r  = mg{L \over 2}\sin 	heta $
Thus, the torque is proportional to sin$	heta$. Hence, option (C) is correct.
$\bullet$ For option (D) : The displacement of midpoint is
${L \over 2} - {L \over 2}\cos 	heta  = {L \over 2}(1 - \cos 	heta )$
That is,
Displacement $\propto$ (1 $-$ cos$	heta$)
Hence, option (D) is correct.

Answer

Instantaneous torque about the point in contact with the floor is proportional to sin$	heta$

Answer

The trajectory of the point A is parabola

Answer

The mid-point of the bar will fall vertically downward

Answer

When the bar makes an angle $	heta$ with the vertical, the displacement of its mid-point from the initial position is proportional to (1 $-$ cos$	heta$)

Centre of Mass and Rotational Motion

A rigid uniform bar AB of length L is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is \theta. Which of the following statements about its motion is/are correct?

Choose the correct answer: