3x Find the vector of the Force when the V = -ye³x + zsin(x) where V is the potential function. If x = t and y = 3t and z = 1 - t, find the moment as a function of time, M(t), and find the angular momentum, H(t), as function of time when the position is 7 = 3î - 2ĵ+ 5k.
3x Find the vector of the Force when the V = -ye³x + zsin(x) where V is the potential function. If x = t and y = 3t and z = 1 - t, find the moment as a function of time, M(t), and find the angular momentum, H(t), as function of time when the position is 7 = 3î - 2ĵ+ 5k.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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5
![**Vector of the Force from Potential Function**
Given the potential function \( V = -ye^{3x} + z \sin(x) \), where \( V \) is the potential function, we aim to find the force vector \(\vec{F}\).
### Expression and Variables
If \( x = t \), \( y = 3t \), and \( z = 1 - t \):
1. **Position vector**:
\[
\vec{r} = 3t\hat{i} - 2t\hat{j} + 5\hat{k}
\]
2. **Moment as a function of time** \( \vec{M}(t) \).
3. **Angular momentum as a function of time** \( \vec{H}(t) \).
### Analytical Steps
1. **Force Calculation**:
The force \(\vec{F}\) is given by the negative gradient of the potential \( V \):
\[
\vec{F} = - \nabla V
\]
To compute this, find the partial derivatives of \( V \).
For \( V = -ye^{3x} + z \sin(x) \):
\[
\frac{\partial V}{\partial x} = -3ye^{3x} + z \cos(x)
\]
\[
\frac{\partial V}{\partial y} = -e^{3x}
\]
\[
\frac{\partial V}{\partial z} = \sin(x)
\]
Therefore, the force vector is:
\[
\vec{F} = - \left( \frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k} \right) = \left( 3ye^{3x} - z \cos(x) \right)\hat{i} + e^{3x}\hat{j} - \sin(x)\hat{k}
\]
2. **Moment Calculation** \( \vec{M}(t) \):
\[
\vec{M}(t) = \vec{r} \times \vec{F}
\]
3. **Angular Momentum Calculation** \( \vec{H}(t)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20dfe5a-a4c1-4793-9f05-a80ad59a67d4%2F2e63985e-8f83-4dc5-830f-164f55947c5b%2Fvfbjlz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vector of the Force from Potential Function**
Given the potential function \( V = -ye^{3x} + z \sin(x) \), where \( V \) is the potential function, we aim to find the force vector \(\vec{F}\).
### Expression and Variables
If \( x = t \), \( y = 3t \), and \( z = 1 - t \):
1. **Position vector**:
\[
\vec{r} = 3t\hat{i} - 2t\hat{j} + 5\hat{k}
\]
2. **Moment as a function of time** \( \vec{M}(t) \).
3. **Angular momentum as a function of time** \( \vec{H}(t) \).
### Analytical Steps
1. **Force Calculation**:
The force \(\vec{F}\) is given by the negative gradient of the potential \( V \):
\[
\vec{F} = - \nabla V
\]
To compute this, find the partial derivatives of \( V \).
For \( V = -ye^{3x} + z \sin(x) \):
\[
\frac{\partial V}{\partial x} = -3ye^{3x} + z \cos(x)
\]
\[
\frac{\partial V}{\partial y} = -e^{3x}
\]
\[
\frac{\partial V}{\partial z} = \sin(x)
\]
Therefore, the force vector is:
\[
\vec{F} = - \left( \frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k} \right) = \left( 3ye^{3x} - z \cos(x) \right)\hat{i} + e^{3x}\hat{j} - \sin(x)\hat{k}
\]
2. **Moment Calculation** \( \vec{M}(t) \):
\[
\vec{M}(t) = \vec{r} \times \vec{F}
\]
3. **Angular Momentum Calculation** \( \vec{H}(t)
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