---
title: "A hollow metal sphere of mass \\(M\\) and volume \\(V\\) is completely submerged and held in place in a tank of liquid with unknown density \\(\\rho\\). The sphere is tethered to the bottom of the tank by two identical, ideal springs, each with a spring constant \\(k\\). In the equilibrium position shown, each spring is stretched by a length \\(\\Delta L\\) from its natural length and makes an angle \\(\\theta\\) with the vertical. Which of the following is a correct expression for the density \\(\\rho\\) of the liquid?"
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url: "https://nerd-notes.com/ubq/109792/"
date_modified: "2026-03-26T05:50:09+00:00"
---

# A hollow metal sphere of mass \(M\) and volume \(V\) is completely submerged and held in place in a tank of liquid with unknown density \(\rho\). The sphere is tethered to the bottom of the tank by two identical, ideal springs, each with a spring constant \(k\). In the equilibrium position shown, each spring is stretched by a length \(\Delta L\) from its natural length and makes an angle \(\theta\) with the vertical. Which of the following is a correct expression for the density \(\rho\) of the liquid?

A hollow metal sphere of mass \(M\) and volume \(V\) is completely submerged and held in place in a tank of liquid with unknown density \(\rho\). The sphere is tethered to the bottom of the tank by two identical, ideal springs, each with a spring constant \(k\). In the equilibrium position shown, each spring is stretched by a length \(\Delta L\) from its natural length and makes an angle \(\theta\) with the vertical. Which of the following is a correct expression for the density \(\rho\) of the liquid?

![A sphere is submerged in a liquid. Two springs are attached to the bottom of the sphere, extending downward and outward to the bottom of the tank. A vertical dashed line passes through the center of the sphere. The angle theta is indicated between this vertical line and each spring. The springs are clearly stretched. Upward force vector labeled F_B and downward force vector labeled Mg are shown on the sphere.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774504209-NYRfg9.jpg)

- **A.** \(\rho = \dfrac{M}{V} + \dfrac{2k \Delta L \cos\theta}{Vg}\)
- **B.** \(\rho = \dfrac{M}{V} - \dfrac{2k \Delta L \cos\theta}{Vg}\)
- **C.** \(\rho = \dfrac{M}{V} + \dfrac{2k \Delta L \sin\theta}{Vg}\)
- **D.** \(\rho = \dfrac{M}{V} + \dfrac{k \Delta L \cos\theta}{Vg}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109792/*