8. The angular momentum L(t) and torque 7(t) of a moving particle with mass m and position vector r(t) are given by L(t) = mr(t) x u(t) and 7(t) = mr(t) x ä(t). Prove that (L(t)) = t(t). What does this say about the angular momentum if there are no dt applied torques (7(t) = 0)? (This is the principal of conservation of angular momentum.) %3D
8. The angular momentum L(t) and torque 7(t) of a moving particle with mass m and position vector r(t) are given by L(t) = mr(t) x u(t) and 7(t) = mr(t) x ä(t). Prove that (L(t)) = t(t). What does this say about the angular momentum if there are no dt applied torques (7(t) = 0)? (This is the principal of conservation of angular momentum.) %3D
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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![**Angular Momentum and Torque in Physics**
In the study of dynamics, particularly concerning a particle with mass \( m \), the concepts of angular momentum \(\vec{L}(t)\) and torque \(\vec{\tau}(t)\) are fundamental. Given the position vector \(\vec{r}(t)\) of the particle, these quantities are defined as:
\[
\vec{L}(t) = m \vec{r}(t) \times \vec{v}(t)
\]
\[
\vec{\tau}(t) = m \vec{r}(t) \times \vec{a}(t)
\]
Where:
- \(\vec{v}(t)\) is the velocity vector.
- \(\vec{a}(t)\) is the acceleration vector.
- \(\times\) denotes the cross product.
**Objective:**
Prove that
\[
\frac{d}{dt}(\vec{L}(t)) = \vec{\tau}(t)
\]
This equation implies the principle of conservation of angular momentum. Specifically, if there are no applied torques \((\vec{\tau}(t) = \vec{0})\), this indicates that the angular momentum of the system remains constant over time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdebf2ee9-b0cd-4a5c-878e-d5c1c95b1f33%2Fada59898-19f0-41c0-a372-5112a65f6948%2F3fu0lmn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Angular Momentum and Torque in Physics**
In the study of dynamics, particularly concerning a particle with mass \( m \), the concepts of angular momentum \(\vec{L}(t)\) and torque \(\vec{\tau}(t)\) are fundamental. Given the position vector \(\vec{r}(t)\) of the particle, these quantities are defined as:
\[
\vec{L}(t) = m \vec{r}(t) \times \vec{v}(t)
\]
\[
\vec{\tau}(t) = m \vec{r}(t) \times \vec{a}(t)
\]
Where:
- \(\vec{v}(t)\) is the velocity vector.
- \(\vec{a}(t)\) is the acceleration vector.
- \(\times\) denotes the cross product.
**Objective:**
Prove that
\[
\frac{d}{dt}(\vec{L}(t)) = \vec{\tau}(t)
\]
This equation implies the principle of conservation of angular momentum. Specifically, if there are no applied torques \((\vec{\tau}(t) = \vec{0})\), this indicates that the angular momentum of the system remains constant over time.
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