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Question

Two rods, one of aluminium and the other of steel, having initial length l1 and l2 respectively are connected together to form a single rod of length l1+l2. The coefficients of linear expansion for aluminium and steel are αa and αs respectively. If the length of each rod increases by the same amount when their temperatures are raised by t ∘C, then find the ratio l1/(l1+l2)

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Solution

The correct option is **C** αs/(αa+αs)

Given:

For aluminium rod,

Initial length =l1

Co-efficient of linear expansion =αa

For steel rod,

Initial length =l2

Co-efficient of linear expansion =αs

To find:

Ratio, l1l1+l2=?

Given, change in length of both the rods are equal when the temperature of both the rods are raised by t ∘C

⇒Δla=Δls

⇒l1αat=l2αst

[ from formula of linear expansion ]

⇒l2l1=αaαs

⇒l2l1+1=αaαs+1

⇒l2+l1l1=αa+αsαs

⇒l1l1+l2=αsαa+αs

Given:

For aluminium rod,

Initial length =l1

Co-efficient of linear expansion =αa

For steel rod,

Initial length =l2

Co-efficient of linear expansion =αs

To find:

Ratio, l1l1+l2=?

Given, change in length of both the rods are equal when the temperature of both the rods are raised by t ∘C

⇒Δla=Δls

⇒l1αat=l2αst

[ from formula of linear expansion ]

⇒l2l1=αaαs

⇒l2l1+1=αaαs+1

⇒l2+l1l1=αa+αsαs

⇒l1l1+l2=αsαa+αs

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