Question

3. The y coordinate of the
centre of mass of the system
of three rods of length 2a'
and two rods of length 'a' as
shown in the figure is:
(Assume all rods to be of
uniform density).Diagram is attached.
​

Answer 1

Answer:

Center of mass of the system of all rods is given as

$x_{cm} = 0$

$y_{cm} = \frac{a3\sqrt3}{16}$

Explanation:

Center of mass of an equilateral triangle will exist only at its centroid

So here we can say that center of mass of the triangle of length 2a will be at

x = 0 and y = $a\frac{\sqrt3}{2} - \frac{a}{\sqrt3} = \frac{a}{2\sqrt3}$

now we have two rods of length a and position of its center of mass is

$y = acos30 - \frac{a}{2}cos30$

$y = \frac{a\sqrt3}{4}$

now for center of mass of complete system of three rods

$y_{cm} = \frac{m_1y_1 + m_2y_2}{m_1 + m_2}$

so we have

$y_{cm} = \frac{6m(\frac{a}{2\sqrt3}) + 2m(\frac{a\sqrt3}{4})}{6m + 2m}$

so we have

$y_{cm} = \frac{a\sqrt3 + a\sqrt3/2}{8}$

$y_{cm} = \frac{a3\sqrt3}{16}$

#Learn

Topic : Center of mass

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