Introduction:
We have a thin bar of length 'l' that has a mass per unit length 'λ'. The mass per unit length increases linearly as we move away from one end, and the increased length has a mass per unit length 'λ''. We need to find the center of mass from the lighter end.

Step 1: Calculating the Total Mass:
To find the center of mass, we first need to calculate the total mass of the bar. The total mass can be obtained by integrating the mass per unit length 'λ' over the entire length of the bar.

Step 2: Determining the Position of the Center of Mass:
Next, we need to determine the position of the center of mass from one end. This can be done by considering the mass distribution along the length of the bar.

Step 3: Setting up the Integral:
Let's assume that the center of mass is located at a distance 'x' from the lighter end of the bar. We can set up an integral to find the position 'x' by considering the mass distribution at different points along the length.

Step 4: Integrating the Mass Distribution:
Integrating the mass distribution involves considering the variable mass per unit length 'λ'. As the mass per unit length increases linearly, we can express it as a function of 'x'. Let's assume the expression for the mass per unit length as 'λ(x)'.

Step 5: Solving the Integral:
By integrating the expression 'λ(x)' over the length of the bar, we can obtain the total mass of the bar. We can then set up an equation to find the position 'x' by equating the total mass to the mass per unit length 'λ'' for the increased length.

Step 6: Calculating the Center of Mass:
Once we have solved the integral and obtained the position 'x', we can calculate the center of mass from the lighter end by subtracting 'x' from the total length 'l'.

Conclusion:
To find the center of mass from the lighter end of a thin bar with a linearly increasing mass per unit length, we need to calculate the total mass of the bar and solve an integral to determine the position 'x'. The center of mass can then be calculated by subtracting 'x' from the total length 'l'.