College Physics, Volume 1
College Physics, Volume 1
2nd Edition
ISBN: 9781133710271
Author: Giordano
Publisher: Cengage
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Chapter 5, Problem 74P
To determine

The ratio of the weight of the twin in the boat to the weight of his brother on solid ground.

Expert Solution & Answer
Check Mark

Answer to Problem 74P

The ratio of the weight of the twin in the boat to the weight of his brother on solid ground is 0.9999_.

Explanation of Solution

College Physics, Volume 1, Chapter 5, Problem 74P

Write the expression for the density of the lake.

    ρlake=mlakeV        (I)

Here, ρlake is the density of lake, mlake is the mass of lake, V is the volume.

Write the expression for the density of Earth.

    ρEarth=mEarthV        (II)

Here, ρEarth is the density of Earth, mEarth is the mass of Earth.

Write the expression for the force due to lake on the twin.

    Flake=Gmlakemtwinr2        (III)

Here, Flake is the force due to the lake acting on twin, G is the universal gravitation constant, mtwin is the mass of twin, r is the distance.

Write the expression for the force due to Earth on the twin.

    FEarth=GmEarthmtwinr2        (IV)

Here, FEarth is the force due to Earth acting on the twin.

Use equation (II) and (I) to solve for the ratio of force of gravity due to the lake on the twin to the Earth acting on the twin.

    FlakeFEarth=Gmlakemtwinr2GmEarthmtwinr2=mlakemEarth=ρlakeρEarth        (V)

Use equation (V) to solve for the FEarth.

    FEarth=(ρlakeρEarth)Flake        (VI)

Write the expression for r.

    r=d2        (VII)

Here, d is the distance between the deepest point on the lake on the twin.

Use equation (VII) to write the expression for the V.

    V=43π(d2)3        (VIII)

Use equation (VIII) in (I) to solve for mlake.

    mlake=ρlake43π(d2)3        (IX)

Use equation (IX) and (VII) in (IV) to solve for FEarth.

    FEarth=Gρlake43π(d2)3mtwin(d2)2=Gρlake43π(d2)mtwin        (X)

Write the expression for the difference in the force due to Earth to the lake acting on the twins.

    ΔF=FEarthFlake        (XI)

Here, ΔF is the difference between the force due to Earth to the lake acting on the twins.

Write the expression for the ratio of total weight experienced by the twins.

    WwaterWEarth=mtwingΔFmtwing        (XII)

Use equation (XI) in (XII) to solve for WwaterWEarth.

    WwaterWEarth=mtwing(FEarthFlake)mtwing        (XIII)

Conclusion:

Substitute 6.67×1011Nm2/kg2 for G, 1000kg/m3 for ρlake, 1600m for d, 70kg for mtwin in equation (X) to find FEarth.

    FEarth=(6.67×1011Nm2/kg2)(1000kg/m3)43π(1600m2)(70kg)=0.016N

Substitute 1000kg/m3 for ρlake, 6000kg/m3 for ρEarth, 0.016N for Flake in equation (VI) to solve for FEarth.

    FEarth=(1000kg/m36000kg/m3)(0.016N)=(16)(0.016N)=0.094N

Substitute 0.094N for FEarth, 0.016N for Flake, 70kg for mtwin, 9.8m/s2 for g in equation (XIII) to find WwaterWEarth.

    WwaterWEarth=(70kg)(9.8m/s2)(0.094N0.016N)(70kg)(9.8m/s2)=0.9999

Therefore, the ratio of the weight of the twin in the boat to the weight of his brother on solid ground is 0.9999_.

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Chapter 5 Solutions

College Physics, Volume 1

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