A solid sphere of mass 8.6 kg, made of metal whose density is $3400 \mathrm {~kg} / \mathrm { m } ^ { 3 }$ hangs 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 which one of the following values?
A) $1900 \mathrm {~kg} / \mathrm { m } ^ { 3 }$
B) $1400 \mathrm {~kg} / \mathrm { m } ^ { 3 }$
C) $1600 \mathrm {~kg} / \mathrm { m } ^ { 3 }$
D) $1700 \mathrm {~kg} / \mathrm { m } ^ { 3 }$
E) $1500 \mathrm {~kg} / \mathrm { m } ^ { 3 }$

Question

A solid sphere of mass  8.6 kg, made of metal whose density is $3400 \mathrm {~kg} / \mathrm { m } ^ { 3 }$ hangs 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 which one of the following values?
A) $1900 \mathrm {~kg} / \mathrm { m } ^ { 3 }$
B) $1400 \mathrm {~kg} / \mathrm { m } ^ { 3 }$
C) $1600 \mathrm {~kg} / \mathrm { m } ^ { 3 }$
D) $1700 \mathrm {~kg} / \mathrm { m } ^ { 3 }$
E) $1500 \mathrm {~kg} / \mathrm { m } ^ { 3 }$

Quizplus · Accepted Answer

The buoyant force acting on the sphere when it is submerged in the liquid can be calculated using the difference between the weight of the sphere in air and the tension in the cord. The weight of the sphere in air is $8.6 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 84.28 \, \text{N}$. The tension in the cord is 38 N, so the buoyant force is $84.28 \, \text{N} - 38 \, \text{N} = 46.28 \, \text{N}$.The volume of the sphere can be found from its mass and the density of the metal: $V = \frac{m}{\rho} = \frac{8.6}{3400} \, \text{m}^3$.The density of the liquid ($\rho_{\text{liquid}}$) can be found using the formula for buoyant force: $F_{\text{buoyant}} = V \times g \times \rho_{\text{liquid}}$, where $V$ is the volume of the sphere, $g$ is the acceleration due to gravity (9.8 m/s$^2$), and $F_{\text{buoyant}}$ is the buoyant force (46.28 N).Solving for $\rho_{\text{liquid}}$, we get $\rho_{\text{liquid}} = \frac{F_{\text{buoyant}}}{V \times g}$.Substituting the values, $\rho_{\text{liquid}} = \frac{46.28}{\frac{8.6}{3400} \times 9.8}$, we find that the density of the liquid is closest to $1900 \, \text{kg/m}^3$.

A Solid Sphere of Mass 8 3400 kg/m33400 \mathrm {~kg} / \mathrm { m } ^ { 3 }3400 kg/m3

A Solid Sphere of Mass 8 $3400 \mathrm {~kg} / \mathrm { m } ^ { 3 }$