Determine the cm of a rod assuming its linear mass density λ (its mass per unit length) varies linearly from λ = λ0 at the left end to double that value, λ = 2λ0

Center Of Mass lies at 5L/9 from O as shown.

Explanation:

The Center of mass of any object is given by

where (x,y) is location of point with mass dm

Now since rod is a linear object we have it's y co-ordinates as zero

Thus

The mass of an element 'dx' at a distance 'x' from O can be written as

Thus we have

Solving with limits form (0,L) we get

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explanation:

we need to know the time interval during which the force acts. impulse of force = momentum change