Ch 14: Fluids and Elasticity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 14: Fluids and Elasticity

문제 23

14장, 문제 23

Styrofoam has a density of 150 kg/m³. What is the maximum mass that can hang without sinking from a 50-cm-diameter Styrofoam sphere in water? Assume the volume of the mass is negligible compared to that of the sphere.

검증된 단계별 안내

Determine the volume of the Styrofoam sphere using the formula for the volume of a sphere:

V = \frac{4}{3} π r^{3}

, where the radius

r

is half the diameter (50 cm = 0.5 m).

Calculate the buoyant force acting on the sphere when it is fully submerged in water using Archimedes' principle:

F = ρ g V

, where

ρ

is the density of water (1000 kg/m³),

g

is the acceleration due to gravity (9.8 m/s²), and

V

is the volume of the sphere.

Determine the weight of the Styrofoam sphere using the formula

W = m g

, where

m

is the mass of the sphere. The mass can be calculated using

m = ρ V

, where

ρ

is the density of Styrofoam (150 kg/m³).

Find the maximum additional mass that can be hung from the sphere without sinking by equating the total weight (weight of the sphere + weight of the additional mass) to the buoyant force. Use the equation:

F = W + m g_{extra}

, where

m_{extra}

is the additional mass.

Solve for the maximum additional mass

m_{extra}

by rearranging the equation:

m_{extra} = \frac{F - W}{g}

. Substitute the values for the buoyant force, weight of the sphere, and gravitational acceleration to find the result.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Density

Density is defined as mass per unit volume and is a crucial property of materials. In this context, the density of Styrofoam (150 kg/m³) indicates how much mass is contained in a given volume. Understanding density helps determine how much mass can be supported by the buoyant force when the Styrofoam sphere is submerged in water.

Buoyancy

Buoyancy is the upward force exerted by a fluid that opposes the weight of an object submerged in it. According to Archimedes' principle, the buoyant force on an object is equal to the weight of the fluid displaced by the object. This concept is essential for calculating the maximum mass that can hang from the Styrofoam sphere without sinking, as it determines how much weight the sphere can support.

Volume of a Sphere

The volume of a sphere is calculated using the formula V = (4/3)πr³, where r is the radius. For a 50-cm-diameter sphere, the radius is 25 cm (0.25 m). Knowing the volume allows us to calculate the weight of the water displaced by the sphere, which is necessary to find the maximum mass that can be suspended without causing the sphere to sink.