/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 68 An open vessel containing water ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 68

An open vessel containing water is given a constant acceleration \(a\) in the horizontal direction. Then the free surface of water gets sloped with the horizontal at an angle \(\theta\) given by (A) \(\theta=\tan ^{-1}\left(\frac{a}{g}\right)\) (B) \(\theta=\tan ^{-1}\left(\frac{g}{a}\right)\) (C) \(\theta=\sin ^{-1}\left(\frac{a}{g}\right)\) (D) \(\theta=\cos ^{-1}\left(\frac{g}{a}\right)\)

Short Answer

Expert verified
The correct answer is (A) \(\theta = \tan^{-1}\left(\frac{a}{g}\right)\), as the angle "胃" between the water surface and the horizontal can be found using the relationship between the horizontal acceleration "a" and the gravitational acceleration "g" and applying trigonometry.

Step by step solution

01

Draw a diagram

First, let's draw a diagram of the problem: Draw an open vessel containing water with a sloped free surface. Label the angle between the free surface and the horizontal as "胃". Mark the direction of the gravitational force "g" and the horizontal acceleration "a".
02

Analysis of the forces involved

On any part of water, the gravitational force is acting vertically downward and the horizontal force due to the acceleration of the vessel is acting horizontally. In order for the water to be in equilibrium, the net inward force on any part of water must be perpendicular to the free surface. This means that the angle "胃" is the angle between the horizontal acceleration "a" and the effective gravitational force acting on the water.
03

Examine the effective gravity and determine its components

The effective gravity is the vector sum of the horizontal acceleration "a" and the gravitational acceleration "g". We can represent these forces as a right-angled triangle. Using trigonometry, we can then relate the angle "胃" to the components of the effective gravity.
04

Apply trigonometry to find the angle "胃"

Using the right-angled triangle formed by the horizontal acceleration "a" and the gravitational acceleration "g", we can apply trigonometry to find the angle "胃". We have: \[\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{g}\] By taking the inverse tangent of both sides, we get: \[ \theta = \tan^{-1}\left(\frac{a}{g}\right) \] Comparing this result to the given options, it is clear that option (A) is the correct answer. Final Answer: (A) \(\theta = \tan^{-1}\left(\frac{a}{g}\right)\)

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