/*! 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 32 Floating Ice Block A floating ic... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 32

Floating Ice Block A floating ice block is pushed through a displacement \(\vec{d}=(15 \mathrm{~m}) \hat{\mathrm{i}}-(12 \mathrm{~m}) \hat{\mathrm{j}}\) along a straight embankment by rushing water, which exerts a force \(\vec{F}=(210 \mathrm{~N}) \mathrm{i}-(150 \mathrm{~N}) \hat{\mathrm{j}}\) on the block. How much work does the force do on the block during the displacement?

Short Answer

Expert verified
4950 J

Step by step solution

01

- Understand the given vectors

Identify the given displacement vector \(\vec{d}\) and the force vector \(\vec{F}\). The displacement vector is \(\vec{d} = (15 \mathrm{~m}) \hat{\mathrm{i}} - (12 \mathrm{~m}) \hat{\mathrm{j}}\), and the force vector is \(\vec{F} = (210 \mathrm{~N}) \hat{\mathrm{i}} - (150 \mathrm{~N}) \hat{\mathrm{j}}\).
02

- Formulate the Work equation

Recall that the work done \((W)\) by a force \(\vec{F}\) on an object during a displacement \(\vec{d}\) is given by the dot product of the force and displacement vectors: \[ W = \vec{F} \cdot \vec{d} \]
03

- Compute the dot product

Calculate the dot product \(\vec{F} \cdot \vec{d}\). The dot product is calculated as follows: \(\vec{F} \cdot \vec{d} = (F_x \cdot d_x) + (F_y \cdot d_y)\). Using the components of \(\vec{F}\) and \(\vec{d}\): \[ \vec{F} \cdot \vec{d} = (210 \mathrm{~N}) \cdot (15 \mathrm{~m}) + (-150 \mathrm{~N}) \cdot (-12 \mathrm{~m}) \] \[ = 3150 \mathrm{~N \cdot m} + 1800 \mathrm{~N \cdot m} \]
04

- Sum the results

Add the two components of the dot product obtained in Step 3: \[ 3150 \mathrm{~N \cdot m} + 1800 \mathrm{~N \cdot m} = 4950 \mathrm{~N \cdot m} = 4950 \mathrm{~J} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Algebra
To fully grasp the problem, understanding vector algebra is essential. Vectors are quantities that have both magnitude and direction. In this problem, both displacement and force are represented as vectors.

The displacement vector \(\vec{d}\) shows how much and in which direction the ice block moves. It is written as \(\vec{d} = (15 \mathrm{~m}) \hat{\mathrm{i}} - (12 \mathrm{~m}) \hat{\mathrm{j}}\), which means the block moves 15 meters to the right (along the x-axis) and 12 meters down (along the y-axis).

The force vector \(\vec{F}\) indicates the direction and magnitude of the force applied by the water. It is written as \(\vec{F} = (210 \mathrm{~N}) \hat{\mathrm{i}} - (150 \mathrm{~N}) \hat{\mathrm{j}}\), meaning a 210 newton force to the right and a 150 newton force downward are acting on the block.

In vector algebra, vectors are often broken down into their components along the x-axis (\

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Most popular questions from this chapter

Large Meteorite vs. TNT On August 10,1972, a large meteorite skipped across the atmosphere above western United States and Canada, much like a stone skipped across water. The accompanying fireball was so bright that it could be seen in the daytime sky (see Fig. \(9-22\) for a similar event). The meteorite's mass was about \(4 \times 10^{6} \mathrm{~kg}\) : its speed was about \(15 \mathrm{~km} / \mathrm{s}\). Had it entered the atmosphere vertically, it would have hit Earth's surface with about the same speed. (a) Calculate the meteorite's loss of kinetic energy (in joules) that would have been associated with the vertical impact. (b) Express the energy as a multiple of the explosive energy of 1 megaton of \(\mathrm{TNT}\), which is \(4.2 \times 10^{15} \mathrm{~J}\). (c) The energy associated with the atomic bomb explosion over Hiroshima was equivalent to 13 kilotons of TNT. To how many Hiroshima bombs would the meteorite impact have been equivalent?

Unmanned Space Probe A \(2500 \mathrm{~kg}\) unmanned space probe is moving in a straight line at a constant speed of \(300 \mathrm{~m} / \mathrm{s}\). Control rockets on the space probe execute a burn in which a thrust of \(3000 \mathrm{~N}\) acts for \(65.0 \mathrm{~s}\). (a) What is the change in the magnitude of the probe's translational momentum if the thrust is backward, forward, or directly sideways? (b) What is the change in kinetic energy under the same three conditions? Assume that the mass of the ejected burn products is negligible compared to the mass of the space probe.

Luge Rider A luge and rider, with a total mass of \(85 \mathrm{~kg}\), emerge from a downhill track onto a horizontal straight track with an initial speed of \(37 \mathrm{~m} / \mathrm{s}\). If they slow at a constant rate of \(2.0 \mathrm{~m} / \mathrm{s}^{2}\) (a) what magnitude \(F\) is required for the slowing force, (b) what distance \(d\) do they travel while slowing, and (c) what work \(W\) is done on them by the slowing force? What are (d) \(F,(\mathrm{e}) d\), and (f) \(W\) if the luge and the rider slow at a rate of \(4.0 \mathrm{~m} / \mathrm{s}^{2} ?\)

Block Dropped on a Spring A \(250 \mathrm{~g}\) block is dropped onto a relaxed vertical spring that has a spring constant of \(k=\) \(2.5 \mathrm{~N} / \mathrm{cm}\) (Fig. \(9-29)\). The block becomes attached to the spring and compresses the spring \(12 \mathrm{~cm}\) before turning around. While the spring is being compressed, what work is done on the block by (a) the gravitational force on it and (b) the spring force? (c) What is the speed of the block just before it hits the spring? (Assume that friction is negligible.) (d) If the speed at impact is doubled, what is the maximum compression of the spring?

Constant Force A constant force of magnitude \(10 \mathrm{~N}\) makes an angle of \(150^{\circ}\) (measured counterclockwise) with the positive \(x\) di- rection as it acts on a \(2.0 \mathrm{~kg}\) object moving in the \(x y\) plane. How much work is done on the object by the force as the object moves from the origin to the point with position vector \((2.0 \mathrm{~m}) \hat{\mathrm{i}}-(4.0 \mathrm{~m}) \hat{\mathrm{j}}\) ?
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