Find the work done in moving a particle from point P(- 5, 0, 2) to Q(3, - 1, 4) if the magnitude and direction of the force are given by v = <7, 3, - 1>.
Find the work done in moving a particle from point P(- 5, 0, 2) to Q(3, - 1, 4) if the magnitude and direction of the force are given by v = <7, 3, - 1>.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Physics Problem: Work Done in Moving a Particle
**Problem Statement:**
Find the work done in moving a particle from point \( P(-5, 0, 2) \) to \( Q(3, -1, 4) \) if the magnitude and direction of the force are given by \( \vec{v} = \langle 7, 3, -1 \rangle \).
**Options:**
1. \( 51 \)
2. \( \langle -5, 22, 31 \rangle \)
3. \( \sqrt{1470} \)
4. \( \sqrt{3149} \)
5. \( 19 \)
**Explanation:**
To find the work done, we use the dot product of the force vector and the displacement vector. The displacement vector \( \vec{d} \) can be calculated as:
\[ \vec{d} = Q - P = \langle 3 - (-5), -1 - 0, 4 - 2 \rangle \]
\[ \vec{d} = \langle 8, -1, 2 \rangle \]
Now, calculate the work done \( W \) using the dot product:
\[ W = \vec{v} \cdot \vec{d} = \langle 7, 3, -1 \rangle \cdot \langle 8, -1, 2 \rangle \]
\[ W = 7 \cdot 8 + 3 \cdot (-1) + (-1) \cdot 2 \]
\[ W = 56 - 3 - 2 \]
\[ W = 51 \]
Therefore, the correct answer is:
\[ \boxed{51} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55815b3c-2575-4201-a9dc-5a4b41a14c1d%2Fb8b45325-fe90-402d-9ce7-0a2dfbc72f4e%2Fnlf30vi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Physics Problem: Work Done in Moving a Particle
**Problem Statement:**
Find the work done in moving a particle from point \( P(-5, 0, 2) \) to \( Q(3, -1, 4) \) if the magnitude and direction of the force are given by \( \vec{v} = \langle 7, 3, -1 \rangle \).
**Options:**
1. \( 51 \)
2. \( \langle -5, 22, 31 \rangle \)
3. \( \sqrt{1470} \)
4. \( \sqrt{3149} \)
5. \( 19 \)
**Explanation:**
To find the work done, we use the dot product of the force vector and the displacement vector. The displacement vector \( \vec{d} \) can be calculated as:
\[ \vec{d} = Q - P = \langle 3 - (-5), -1 - 0, 4 - 2 \rangle \]
\[ \vec{d} = \langle 8, -1, 2 \rangle \]
Now, calculate the work done \( W \) using the dot product:
\[ W = \vec{v} \cdot \vec{d} = \langle 7, 3, -1 \rangle \cdot \langle 8, -1, 2 \rangle \]
\[ W = 7 \cdot 8 + 3 \cdot (-1) + (-1) \cdot 2 \]
\[ W = 56 - 3 - 2 \]
\[ W = 51 \]
Therefore, the correct answer is:
\[ \boxed{51} \]
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