Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Question
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Chapter 13, Problem 110P

(a)

To determine

Find the relation for the radius of the metallic strips.

(a)

Expert Solution
Check Mark

Answer to Problem 110P

Thus proved that the radius of the metallic strips is Rs(α2α1)ΔT.

Explanation of Solution

Write the equation for linear expansion of the one metallic strip.

ΔL1L0=α1ΔT (I)

Here, ΔL1 is the change in length of the first metallic strip, L0 is the original length, α1 is the coefficient of linear expansion of the first metallic strip, and ΔT is the change in temperature.

Write the equation for linear expansion of the metallic strip.

ΔL2L0=α2ΔT (II)

Here, ΔL2 is the change in length of the second metallic strip, L0 is the original length, α2 is the coefficient of linear expansion of the second metallic strip, and ΔT is the change in temperature.

Rewrite the equation (I) for ΔL1.

ΔL1=L0α1ΔT (III)

Rewrite the equation (II) for ΔL2.

ΔL2=L0α2ΔT (IV)

If the expansion coefficient of the bimetallic coefficient α2>α1.

The relation for the radius of the one strip is.

L0+ΔL1=Rθ (V)

Here, R is the radius of the circular arc and θ is the arc length

The relation for the radius of another strip is.

L0+ΔL2=(R+s)θ (VI)

Here, s is the thickness of the layer

Rewrite the equation (V) for θ.

θ=L0+ΔL1R (VII)

Conclusion:

Substitute equation (VII) in equation (VI).

L0+ΔL2=(R+s)L0+ΔL1R

Substitute the equation (III) and (IV) in above equation.

R(L0+L0α2ΔT)=(R+s)(L0+L0α1ΔT)L0R(1+α2ΔT)=RL0(1+α1ΔT)+sL0(1+α1ΔT)R(1+α2ΔT)=R(1+α1ΔT)+s(1+α1ΔT)R(1+α2ΔT)R(1+α1ΔT)=s(1+α1ΔT)

Apply the condition αΔT1 and solve the above equation for R.

R(α2α1)ΔT=s(1+α1ΔT)R(α2α1)ΔT=sRs(α2α1)ΔT

Hence, proved the radius of the metallic strips is Rs(α2α1)ΔT.

(b)

To determine

The radius of the bimetallic strip.

(b)

Expert Solution
Check Mark

Answer to Problem 110P

The radius of the bimetallic strip is 0.7m.

Explanation of Solution

Write the equation for radius of the bimetallic strip from part (a).

R=s(αbrassαiron)ΔT (VIII)

Here, R is the radius of the bimetallic strip, s is the thickness of the each layer, αiron is the coefficient of linear expansion of the iron, αbrass is the coefficient of linear expansion of the brass, and ΔT is the change in temperature.

Conclusion:

Substitute 0.1mm for s, 19×106K1 for αbrass, 12×106K1 for αiron, and 20.0°C for ΔT in equation (VIII).

R=(0.1mm)(103m1mm)[(1912)×106K1]20.0K=7.1×101m=0.71m0.7m

Therefore, the radius of the bimetallic strip is 0.7m.

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Chapter 13 Solutions

Physics

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