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Sagot :
Sure, let's solve the given problem step by step:
### Problem:
A particle moves from a point [tex]\((35, -4, -2)\)[/tex] to another point [tex]\((5, -6, 2)\)[/tex] under the influence of a force [tex]\(\vec{F} = -3\hat{i} + 4\hat{j} + 4\hat{k} \, \text{N}\)[/tex]. Calculate the work done by the force.
### Solution:
1. Identify Initial and Final Points:
We have the initial point [tex]\((35, -4, -2)\)[/tex] and the final point [tex]\((5, -6, 2)\)[/tex].
2. Determine the Displacement Vector:
The displacement vector [tex]\(\vec{d}\)[/tex] can be found by subtracting the coordinates of the initial point from the coordinates of the final point:
[tex]\[ \vec{d} = \text{final point} - \text{initial point} \][/tex]
Given:
[tex]\[ \text{initial point} = (35, -4, -2) \][/tex]
[tex]\[ \text{final point} = (5, -6, 2) \][/tex]
Therefore:
[tex]\[ \vec{d} = (5 - 35, -6 + 4, 2 + 2) \][/tex]
Simplifying:
[tex]\[ \vec{d} = (-30, -2, 4) \][/tex]
Thus, the displacement vector [tex]\(\vec{d} = -30\hat{i} - 2\hat{j} + 4\hat{k}\)[/tex].
3. Force Vector:
Given the force vector:
[tex]\[ \vec{F} = -3\hat{i} + 4\hat{j} + 4\hat{k} \, \text{N} \][/tex]
4. Calculate the Work Done:
Work done by a force [tex]\(\vec{F}\)[/tex] over a displacement [tex]\(\vec{d}\)[/tex] is given by the dot product of [tex]\(\vec{F}\)[/tex] and [tex]\(\vec{d}\)[/tex]:
[tex]\[ W = \vec{F} \cdot \vec{d} \][/tex]
Substituting the given vectors:
[tex]\[ \vec{F} = (-3, 4, 4) \][/tex]
[tex]\[ \vec{d} = (-30, -2, 4) \][/tex]
The dot product [tex]\(\vec{F} \cdot \vec{d}\)[/tex] is:
[tex]\[ W = (-3 \times -30) + (4 \times -2) + (4 \times 4) \][/tex]
Simplifying each term:
[tex]\[ W = 90 + (-8) + 16 \][/tex]
Summing these:
[tex]\[ W = 90 - 8 + 16 \][/tex]
[tex]\[ W = 98 \, \text{J} \][/tex]
Therefore, the work done by the force is [tex]\(98 \, \text{J}\)[/tex].
### Problem:
A particle moves from a point [tex]\((35, -4, -2)\)[/tex] to another point [tex]\((5, -6, 2)\)[/tex] under the influence of a force [tex]\(\vec{F} = -3\hat{i} + 4\hat{j} + 4\hat{k} \, \text{N}\)[/tex]. Calculate the work done by the force.
### Solution:
1. Identify Initial and Final Points:
We have the initial point [tex]\((35, -4, -2)\)[/tex] and the final point [tex]\((5, -6, 2)\)[/tex].
2. Determine the Displacement Vector:
The displacement vector [tex]\(\vec{d}\)[/tex] can be found by subtracting the coordinates of the initial point from the coordinates of the final point:
[tex]\[ \vec{d} = \text{final point} - \text{initial point} \][/tex]
Given:
[tex]\[ \text{initial point} = (35, -4, -2) \][/tex]
[tex]\[ \text{final point} = (5, -6, 2) \][/tex]
Therefore:
[tex]\[ \vec{d} = (5 - 35, -6 + 4, 2 + 2) \][/tex]
Simplifying:
[tex]\[ \vec{d} = (-30, -2, 4) \][/tex]
Thus, the displacement vector [tex]\(\vec{d} = -30\hat{i} - 2\hat{j} + 4\hat{k}\)[/tex].
3. Force Vector:
Given the force vector:
[tex]\[ \vec{F} = -3\hat{i} + 4\hat{j} + 4\hat{k} \, \text{N} \][/tex]
4. Calculate the Work Done:
Work done by a force [tex]\(\vec{F}\)[/tex] over a displacement [tex]\(\vec{d}\)[/tex] is given by the dot product of [tex]\(\vec{F}\)[/tex] and [tex]\(\vec{d}\)[/tex]:
[tex]\[ W = \vec{F} \cdot \vec{d} \][/tex]
Substituting the given vectors:
[tex]\[ \vec{F} = (-3, 4, 4) \][/tex]
[tex]\[ \vec{d} = (-30, -2, 4) \][/tex]
The dot product [tex]\(\vec{F} \cdot \vec{d}\)[/tex] is:
[tex]\[ W = (-3 \times -30) + (4 \times -2) + (4 \times 4) \][/tex]
Simplifying each term:
[tex]\[ W = 90 + (-8) + 16 \][/tex]
Summing these:
[tex]\[ W = 90 - 8 + 16 \][/tex]
[tex]\[ W = 98 \, \text{J} \][/tex]
Therefore, the work done by the force is [tex]\(98 \, \text{J}\)[/tex].
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