An iron rod of length 50 cm is joined at an end to an aluminium rod of length 100 cm. All measurements are made at 20∘C. Find the length of the composite system at 100∘C and its average coefficient of linear expansion. The coefficient of linear expansion of iron and aluminium are 12 × 10−6/∘C and 24 × 10−6/∘C respectively.
An iron rod of length 50 cm is joined at an end to an aluminium rod of length 100 cm. All measurements are made at 20∘C. Find the length of the composite system at 100∘C and its average coefficient of linear expansion. The coefficient of linear expansion of iron and aluminium are 12 × 10−6/∘C and 24 × 10−6/∘C respectively.
The correct option is C 20 × 10−6/∘C
The length of the iron rod at 100∘C -
l1 = (50cm)[1 + (12 × 10−6/∘C)(100∘C−20∘C)]
= 50.048 cm.
The length of the aluminium rod at 100∘C -
l2 = (100cm)[1 + (24 × 10−6/∘C)(100∘C−20∘C)]
= 100.192 cm.
The length of the composite system at 100∘C is - (50.048 cm + 100.192 cm) = 150.24 cm.
But the length of the composite system at 20∘C is 150 cm.
So, the average coefficient of linear expansion of the composite rod is
a = 0.24cm150 cm × (100∘C−20∘C)
= 20 × 10−6/∘C.
The length of the iron rod at 100∘C -
l1 = (50cm)[1 + (12 × 10−6/∘C)(100∘C−20∘C)]
= 50.048 cm.
The length of the aluminium rod at 100∘C -
l2 = (100cm)[1 + (24 × 10−6/∘C)(100∘C−20∘C)]
= 100.192 cm.
The length of the composite system at 100∘C is - (50.048 cm + 100.192 cm) = 150.24 cm.
But the length of the composite system at 20∘C is 150 cm.
So, the average coefficient of linear expansion of the composite rod is
a = 0.24cm150 cm × (100∘C−20∘C)
= 20 × 10−6/∘C.
