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Question

A steel scale is correct at 0oC, the length of a brass tube measured by it at 40oC is 5m. The correct length of the tube at 0oC is (Coefficients of linear expansion of steel and brass are 11×106/oC and 19×106/oC respectively).
  1. 4.001m
  2. 4.999m
  3. 4.501m
  4. 5.001m

A
4.001m
B
5.001m
C
4.999m
D
4.501m
Solution
Verified by Toppr

ls=lb=5 m

Tf=400c

Actual length of brass tube, lb=?

Ti=0

For Length of scale at 00=ls, ls=ls(1+αsΔT)

5=ls(1+40αs)

ls=51+40αs


Actual length of tube at 400

lb=ls+Δls

=ls+lsls
=2lsls

lb=(2×551+40αs)

Actual length of tube at 00c=lb

lb=lb(1+α04T)

lb=lb1+αb4T=(1051+40αs)×11+40αb

=(1051+40αs)×11+40αb

=(1051+40×11×106)×(11+40×19×106)

=(1051+440×106)×(11+760×106)

=(1051.000440)×(11+00076)

=4.9984.999m

or

Let the length of tube and scale at 0oC be l. Then,
Change in measured tube length
Δl=Thermal expansion of brass Thermal expansion of steel
Δl=lαBΔTlΔαBΔT
Δl=lΔT(αBαS)
Δl=l(400)(19×10611×106)
So, new length measured by scale=l+Δl
5=l+l(40)(7×106)
51+40×7×106=l
l=4.99864.999m
Option C is correct.

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