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(a) Suppose a meter stick made of steel and one made of invar (an alloy of iron and nickel) are the same length at 0ยบC . What is their difference in length at 22.0ยบC ? (b) Repeat the calculation for two 30.0-m-long surveyorโ€™s tapes.

Short Answer

Expert verified

(a) The difference in length of meter stick made of steel and invar is 2.44ร—10-4m for 1 metre of length, and

(b) for 30 meters of length.

Step by step solution

01

Introduction

We calculate the change in length of the steel and invar stick by the formula for linear expansion in solids and find the difference in lengths.

02

Given parameters and formula for linear expansion

Coefficient of linear expansion of steel =12ร—10-6/ยฐC12ร—10-6/ยฐC

Formula for length expansion of solids =โˆ†L=ฮฑLโˆ†T

Here,

ฮฑ- coefficient of linear expansion.

L-original length

โˆ†T- change in temperature.

03

 Increase in length for steel & invar  of length 1 metre

(a)

For steel

โˆ†L=ฮฑLโˆ†T=(1.2ร—10-5/ยฐC)ร—1mร—(22ยฐC)=2.64ร—10-4m

For invar

โˆ†L=ฮฑLโˆ†T=(9.0ร—10-6/ยฐC)ร—1mร—(22ยฐC)=1.98ร—10-5m

Differenceinlength=(2.64ร—10-4m)-(1.98ร—10-5m)=2.44ร—10-4m

04

 Increase in length for steel & invar  of length 30 metres

(b)

For steel

โˆ†L=ฮฑLโˆ†T=(1.2ร—10-5/ยฐC)ร—30mร—(22ยฐC)=7.92ร—10-3m

For invar

โˆ†L=ฮฑLโˆ†T=(9.0ร—10-6/ยฐC)ร—30mร—(22ยฐC)=5.94ร—10-4m

Differenceinlenght=(7.92ร—10-3m)-(5.94ร—10-4m)=7.33ร—10-3m(1mm10-3m)=7.33mm

The difference in length of meter stick made of steel and invar is 2.44 X 10-4 m for 1 metre of length and7.33 mmfor 30 meters of length.

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