The mass per unit length of a rod varies as λ = (M₀/L)x where M₀ is a constant and x is the distance of any point on rod measured from one end. Centre of mass of the rod from the given end is at a distance of KL/3 on the rod. Find the value of K.
Physics · Mixed Chapter
JEE Mains

Question

The mass per unit length of a rod varies as λ = (M₀/L)x where M₀ is a constant and x is the distance of any point on rod measured from one end. Centre of mass of the rod from the given end is at a distance of KL/3 on the rod. Find the value of K.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: The linear mass density is λ(x)=(M₀/L)x. Total mass is Mtot=∫₀ᴸλ dx=(M₀/L)(L²/2). Step 2: First moment about the given end is ∫₀ᴸx λ(x)dx=(M₀/L)(L³/3). Step 3: Centre of mass x꜀ₘ=(first moment)/(total mass)=(L³/3)/(L²/2)=2L/3. Step 4: Since x꜀ₘ=KL/3, K=2.