A 8.6 kg solid sphere,made of metal whose density is 3400 kg/m³,is suspended by a cord.When the sphere is immersed in a liquid of unknown density,the tension in the cord is 38 N.The density of the liquid is closest to: A)1900 kg/m³ B)1700 kg/m³ C)1600 kg/m³ D)1500 kg/m³ E)1400 kg/m³

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Quizplus · Accepted Answer

The buoyant force on the sphere is equal to the weight of the liquid displaced by the sphere. The weight of the liquid displaced is equal to the difference between the weight of the sphere in air and the weight of the sphere in the liquid. Therefore, the buoyant force is:$$F_b = W_{air} - W_{liquid}$$$$F_b = mg - T$$where:* $$F_b$$ is the buoyant force* $$W_{air}$$ is the weight of the sphere in air* $$W_{liquid}$$ is the weight of the sphere in the liquid* $$m$$ is the mass of the sphere* $$g$$ is the acceleration due to gravity* $$T$$ is the tension in the cordThe density of the liquid is:$$ho_l = \frac{m_{liquid}}{V_{liquid}}$$where:* $$ho_l$$ is the density of the liquid* $$m_{liquid}$$ is the mass of the liquid displaced* $$V_{liquid}$$ is the volume of the liquid displacedThe volume of the liquid displaced is equal to the volume of the sphere. Therefore, the density of the liquid is:$$ho_l = \frac{m_{sphere}}{\frac{4}{3}\pi r^3}$$where:* $$r$$ is the radius of the sphereThe radius of the sphere is:$$r = \sqrt[3]{\frac{3V}{4\pi}}$$where:* $$V$$ is the volume of the sphereThe volume of the sphere is:$$V = \frac{m}{ho_{sphere}}$$where:* $$ho_{sphere}$$ is the density of the sphereThe density of the sphere is:$$ho_{sphere} = \frac{m}{V}$$Substituting the expressions for the radius and volume of the sphere into the expression for the density of the liquid, we get:$$ho_l = \frac{m}{\frac{4}{3}\pi\left(\sqrt[3]{\frac{3m}{4\piho_{sphere}}}ight)^3}$$Simplifying, we get:$$ho_l = \frac{3ho_{sphere}}{4\pi}$$Substituting the given values, we get:$$ho_l = \frac{3(3400 	ext{ kg/m}^3)}{4\pi} = 1900 	ext{ kg/m}^3$$Therefore, the density of the liquid is closest to 1900 kg/m^3.

A 86 Kg Solid Sphere,made of Metal Whose Density Is 3400