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title: "A block of mass \\(m\\) and density \\(\\rho_b\\) is compressed a distance \\(x\\) against a spring of constant \\(k\\) at the base of a ramp. The block is released from rest and slides up the ramp of length \\(L\\) inclined at an angle \\(\\theta\\) above the horizontal, where the coefficient of kinetic friction between the block and the ramp is \\(\\mu\\). After leaving the ramp, the block travels through the air and falls a vertical distance \\(H\\) before entering a pool of water of density \\(\\rho_w\\). The block comes to rest at a depth \\(d\\) below the water surface. An average resistive drag force \\(F_R\\) acts on the block while it is in the water. Which of the following expressions correctly relates the initial energy stored in the spring to the final state of the block?"
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url: "https://nerd-notes.com/ubq/109879/"
date_modified: "2026-04-04T01:35:27+00:00"
---

# A block of mass \(m\) and density \(\rho_b\) is compressed a distance \(x\) against a spring of constant \(k\) at the base of a ramp. The block is released from rest and slides up the ramp of length \(L\) inclined at an angle \(\theta\) above the horizontal, where the coefficient of kinetic friction between the block and the ramp is \(\mu\). After leaving the ramp, the block travels through the air and falls a vertical distance \(H\) before entering a pool of water of density \(\rho_w\). The block comes to rest at a depth \(d\) below the water surface. An average resistive drag force \(F_R\) acts on the block while it is in the water. Which of the following expressions correctly relates the initial energy stored in the spring to the final state of the block?

A block of mass \(m\) and density \(\rho_b\) is compressed a distance \(x\) against a spring of constant \(k\) at the base of a ramp. The block is released from rest and slides up the ramp of length \(L\) inclined at an angle \(\theta\) above the horizontal, where the coefficient of kinetic friction between the block and the ramp is \(\mu\). After leaving the ramp, the block travels through the air and falls a vertical distance \(H\) before entering a pool of water of density \(\rho_w\). The block comes to rest at a depth \(d\) below the water surface. An average resistive drag force \(F_R\) acts on the block while it is in the water. Which of the following expressions correctly relates the initial energy stored in the spring to the final state of the block?

![A schematic showing the path of a block of mass m. At the start (point 1), the block is compressed against a horizontal spring k by distance x at the bottom of a ramp. The ramp has length L and is at an angle theta. After the ramp, the block follows a parabolic path through the air. The vertical height from the ramp's end to the surface of a water tank is H. The block is then shown submerged in the water at a depth d (point 2).](https://nerd-notes.com/wp-content/uploads/ubq-diagrams/ubq-frq-generatedstem-fig-1-1775266526-2urEp8.jpg)

- **A.** \( \dfrac{1}{2}kx^2 = mg(L\sin\theta - H - d) + \mu mgL\cos\theta + F_R d + \dfrac{\rho_w}{\rho_b}mgd \)
- **B.** \( \dfrac{1}{2}kx^2 = mg(L\sin\theta - H - d) + \mu mgL\sin\theta + F_R d + \dfrac{\rho_w}{\rho_b}mgd \)
- **C.** \( \dfrac{1}{2}kx^2 = mg(L\sin\theta + H + d) + \mu mgL\cos\theta + F_R d + \dfrac{\rho_w}{\rho_b}mgd \)
- **D.** \( \dfrac{1}{2}kx^2 = mg(L\sin\theta - H - d) + \mu mgL\cos\theta + F_R d - \dfrac{\rho_w}{\rho_b}mgd \)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109879/*