rotational vectors and angular momentum 298 summer... · summer 2018 prof. sergio b. mendes 2...
TRANSCRIPT
![Page 1: Rotational Vectors and Angular Momentum 298 summer... · Summer 2018 Prof. Sergio B. Mendes 2 Angular Velocity actually needs to be described by a vector: • Angular velocity is](https://reader034.vdocuments.us/reader034/viewer/2022042303/5ece26e2cebd7c0f84040f8b/html5/thumbnails/1.jpg)
1Prof. Sergio B. MendesSummer 2018
Chapter 11 of Essential University Physics, Richard Wolfson, 3rd Edition
Rotational Vectors and Angular Momentum
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2Prof. Sergio B. MendesSummer 2018
Angular Velocity actually needs to be described by a vector:
• Angular velocity is a vector 𝝎𝝎 along the axis of rotation
• Direction of the vector 𝝎𝝎 is determined by the right-hand rule
• Magnitude of the vector 𝝎𝝎 is determined by 𝜔𝜔 = 𝑑𝑑𝜃𝜃𝑑𝑑𝑑𝑑
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3Prof. Sergio B. MendesSummer 2018
Angular Acceleration is also a vector: ��𝛼 ≡ 𝑑𝑑𝜔𝜔
𝑑𝑑𝑑𝑑
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4Prof. Sergio B. MendesSummer 2018
A Different Kind of Product Between Two Vectors
A Math Break:
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5Prof. Sergio B. MendesSummer 2018
Definition of Cross Product:
𝑨𝑨 × 𝑩𝑩
𝑨𝑨 × 𝑩𝑩 = −𝑩𝑩 × 𝑨𝑨
𝑨𝑨 × 𝑩𝑩 + 𝑪𝑪 = 𝑨𝑨 × 𝑩𝑩 + 𝑨𝑨 × 𝑪𝑪
P1:
P2:
𝑨𝑨
𝑩𝑩𝜃𝜃
𝑪𝑪
• Direction is determined by the right-hand rule
• Magnitude is determined by 𝑪𝑪 = 𝐶𝐶 = 𝐴𝐴 𝐵𝐵 sin 𝜃𝜃
𝑪𝑪 ⊥ 𝑨𝑨
𝑪𝑪 ⊥ 𝑩𝑩≡ 𝑪𝑪
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6Prof. Sergio B. MendesSummer 2018
��𝒊 × 𝒋𝒋 = �𝒌𝒌
𝑥𝑥
𝑦𝑦
𝑧𝑧
��𝒊��𝒋
�𝒌𝒌
�𝒌𝒌 × ��𝒊 = 𝒋𝒋
𝒋𝒋 × �𝒌𝒌 = ��𝒊 ��𝒊 × ��𝒊 = 𝒋𝒋 × 𝒋𝒋 = �𝒌𝒌 × �𝒌𝒌 = 𝟎𝟎
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7Prof. Sergio B. MendesSummer 2018
= 𝐴𝐴𝑦𝑦 𝐵𝐵𝑧𝑧 − 𝐴𝐴𝑧𝑧 𝐵𝐵𝑦𝑦 ��𝒊 + 𝐴𝐴𝑧𝑧 𝐵𝐵𝑥𝑥 − 𝐴𝐴𝑥𝑥 𝐵𝐵𝑧𝑧 𝒋𝒋 + 𝐴𝐴𝑥𝑥 𝐵𝐵𝑦𝑦 − 𝐴𝐴𝑦𝑦 𝐵𝐵𝑥𝑥 �𝒌𝒌
𝑨𝑨 = 𝐴𝐴𝑥𝑥 ��𝒊 + 𝐴𝐴𝑦𝑦 𝒋𝒋 + 𝐴𝐴𝑧𝑧 �𝒌𝒌
𝑩𝑩 = 𝐵𝐵𝑥𝑥 ��𝒊 + 𝐵𝐵𝑦𝑦 𝒋𝒋 + 𝐵𝐵𝑧𝑧 �𝒌𝒌
𝑨𝑨 × 𝑩𝑩 = 𝐴𝐴𝑥𝑥 ��𝒊 + 𝐴𝐴𝑦𝑦 𝒋𝒋 + 𝐴𝐴𝑧𝑧 �𝒌𝒌 × 𝐵𝐵𝑥𝑥 ��𝒊 + 𝐵𝐵𝑦𝑦 𝒋𝒋 + 𝐵𝐵𝑧𝑧 �𝒌𝒌
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8Prof. Sergio B. MendesSummer 2018
Back to the Physics:
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9Prof. Sergio B. MendesSummer 2018
Torque is the cross product of the two vectors: 𝒓𝒓 and 𝑭𝑭
𝝉𝝉 ≡ 𝒓𝒓 × 𝑭𝑭
𝒓𝒓
𝑭𝑭𝜃𝜃
𝝉𝝉
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10Prof. Sergio B. MendesSummer 2018
Let’s see if this definition satisfies what we learned so far:
𝝉𝝉 ≡ 𝒓𝒓 × 𝑭𝑭
𝜏𝜏 = 𝑟𝑟 𝐹𝐹 sin 𝜃𝜃
𝜏𝜏 ∥ 𝛼𝛼
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11Prof. Sergio B. MendesSummer 2018
Definition of Angular Momentum:
𝑳𝑳 ≡ 𝒓𝒓 × 𝒑𝒑
𝒓𝒓
𝒑𝒑𝜃𝜃
𝑳𝑳
𝑳𝑳 ⊥ 𝒓𝒓
𝑳𝑳 ⊥ 𝒑𝒑
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12Prof. Sergio B. MendesSummer 2018
Example 11.1
𝑳𝑳 ≡ 𝒓𝒓 × 𝒑𝒑
𝒓𝒓 ⊥ 𝒗𝒗
𝒓𝒓 ⊥ 𝒑𝒑 = 𝑚𝑚 𝒗𝒗
𝐿𝐿 = 𝑟𝑟 𝑝𝑝 𝑠𝑠𝑠𝑠𝑠𝑠 90°= 𝑟𝑟 𝑝𝑝= 𝑟𝑟 𝑚𝑚 𝑣𝑣= 𝑟𝑟 𝑚𝑚 𝜔𝜔 𝑟𝑟
= 𝑚𝑚 𝑟𝑟2 𝜔𝜔
𝑳𝑳 = 𝐿𝐿𝑥𝑥 ��𝒊 + 𝐿𝐿𝑦𝑦 𝒋𝒋 + 𝐿𝐿𝑧𝑧 �𝒌𝒌
= 𝐿𝐿𝑧𝑧 �𝒌𝒌
= 𝑚𝑚 𝑟𝑟2 𝜔𝜔 �𝒌𝒌
= 𝑚𝑚 𝑟𝑟2 𝝎𝝎= 𝐼𝐼 𝝎𝝎
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13Prof. Sergio B. MendesSummer 2018
In linear motion we had:
𝒑𝒑 = 𝑚𝑚 𝒗𝒗
now, in rotational motion we have:
𝑳𝑳 = 𝐼𝐼 𝝎𝝎
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14Prof. Sergio B. MendesSummer 2018
We learned that Newton’s Second Law can be written in terms of the
linear momentum 𝒑𝒑 ≡ 𝑚𝑚 𝒗𝒗 as:𝑭𝑭 = 𝑚𝑚 𝒂𝒂
= 𝑚𝑚𝑑𝑑𝒗𝒗𝑑𝑑𝑑𝑑
=𝑑𝑑 𝑚𝑚 𝒗𝒗𝑑𝑑𝑑𝑑
=𝑑𝑑𝒑𝒑𝑑𝑑𝑑𝑑
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15Prof. Sergio B. MendesSummer 2018
= �𝑳𝑳𝒊𝒊
Now, the connection between Angular Momentum 𝑳𝑳 and Torque 𝝉𝝉 :
= � 𝒓𝒓𝒊𝒊 × 𝒑𝒑𝒊𝒊
𝑑𝑑𝑳𝑳𝑑𝑑𝑑𝑑
=𝑑𝑑𝑑𝑑𝑑𝑑
�𝑳𝑳𝒊𝒊 =𝑑𝑑𝑑𝑑𝑑𝑑
� 𝒓𝒓𝒊𝒊 × 𝒑𝒑𝒊𝒊
= �𝝉𝝉𝒊𝒊 = 𝝉𝝉𝒏𝒏𝒏𝒏𝒏𝒏,𝒏𝒏𝒆𝒆𝒏𝒏
𝑳𝑳
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16Prof. Sergio B. MendesSummer 2018
In linear motion we had
𝑭𝑭𝒏𝒏𝒏𝒏𝒏𝒏,𝒏𝒏𝒆𝒆𝒏𝒏 =𝑑𝑑𝑷𝑷𝑑𝑑𝑑𝑑
now, in rotational motion we have:
𝝉𝝉𝒏𝒏𝒏𝒏𝒏𝒏,𝒏𝒏𝒆𝒆𝒏𝒏 =𝑑𝑑𝑳𝑳𝑑𝑑𝑑𝑑
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17Prof. Sergio B. MendesSummer 2018
𝑳𝑳 = 𝐼𝐼 𝝎𝝎 = 𝒄𝒄𝒄𝒄𝒏𝒏𝒄𝒄𝒏𝒏𝒂𝒂𝒏𝒏𝒏𝒏𝝉𝝉𝒏𝒏𝒏𝒏𝒏𝒏,𝒏𝒏𝒆𝒆𝒏𝒏 = 0
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19Prof. Sergio B. MendesSummer 2018
Example 11.2𝜔𝜔1 = 45 days per revolution
𝑅𝑅1 = 20 Mm
𝑅𝑅2 = 6 km
𝜔𝜔2 = ? ?
𝐼𝐼1 𝜔𝜔1 = 𝐼𝐼2 𝜔𝜔2
𝐼𝐼 =25𝑀𝑀 𝑅𝑅2
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20Prof. Sergio B. MendesSummer 2018
On the playground
𝐿𝐿𝑏𝑏, 𝑖𝑖𝑖𝑖 = 0
𝐿𝐿𝑚𝑚𝑚𝑚𝑚𝑚, 𝑖𝑖𝑖𝑖 = 𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝑖𝑖𝑖𝑖
𝐿𝐿𝑚𝑚, 𝑖𝑖𝑖𝑖 = 𝑅𝑅 𝑚𝑚𝑚𝑚 𝑣𝑣𝑚𝑚
𝐿𝐿𝑚𝑚𝑚𝑚𝑚𝑚, 𝑓𝑓𝑖𝑖 = 𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝑓𝑓𝑖𝑖
𝐿𝐿𝑏𝑏, 𝑓𝑓𝑖𝑖 = 𝐼𝐼𝑏𝑏 𝜔𝜔𝑓𝑓𝑖𝑖
𝐿𝐿𝑚𝑚, 𝑓𝑓𝑖𝑖 = 𝐼𝐼𝑚𝑚 𝜔𝜔𝑓𝑓𝑖𝑖
𝐿𝐿𝑚𝑚𝑚𝑚𝑚𝑚, 𝑖𝑖𝑖𝑖 + 𝐿𝐿𝑏𝑏, 𝑖𝑖𝑖𝑖 + 𝐿𝐿𝑚𝑚, 𝑖𝑖𝑖𝑖 = 𝐿𝐿𝑚𝑚𝑚𝑚𝑚𝑚, 𝑓𝑓𝑖𝑖 + 𝐿𝐿𝑏𝑏, 𝑓𝑓𝑖𝑖 + 𝐿𝐿𝑚𝑚, 𝑓𝑓𝑖𝑖
𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝑖𝑖𝑖𝑖 + 𝑅𝑅 𝑚𝑚𝑚𝑚 𝑣𝑣𝑚𝑚 = 𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔𝑓𝑓𝑖𝑖 + 𝐼𝐼𝑏𝑏 𝜔𝜔𝑓𝑓𝑖𝑖 + 𝐼𝐼𝑚𝑚 𝜔𝜔𝑓𝑓𝑖𝑖
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21Prof. Sergio B. MendesSummer 2018
𝑳𝑳 = 𝐼𝐼 𝝎𝝎 = 𝒄𝒄𝒄𝒄𝒏𝒏𝒄𝒄𝒏𝒏𝒂𝒂𝒏𝒏𝒏𝒏𝝉𝝉𝒏𝒏𝒏𝒏𝒏𝒏,𝒏𝒏𝒆𝒆𝒏𝒏 = 0
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23Prof. Sergio B. MendesSummer 2018
Gyroscopes
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25Prof. Sergio B. MendesSummer 2018
Gyroscope Precession
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26Prof. Sergio B. MendesSummer 2018
Addendum to Chapter 8 about Gravitation
𝑭𝑭 = − 𝐺𝐺𝑀𝑀 𝑚𝑚𝑟𝑟2
��𝑟
𝝉𝝉 = 𝒓𝒓 × 𝑭𝑭 = 0
𝑳𝑳 = 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑑𝑑𝑐𝑐𝑠𝑠𝑑𝑑 = 𝒓𝒓 × 𝑚𝑚 𝒗𝒗
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27Prof. Sergio B. MendesSummer 2018
𝒓𝒓 = 𝑟𝑟 �𝒓𝒓
𝒗𝒗 = �𝒓𝒓𝑑𝑑𝑟𝑟𝑑𝑑𝑑𝑑
+ �𝜽𝜽 𝑟𝑟𝑑𝑑𝜃𝜃𝑑𝑑𝑑𝑑
𝑳𝑳 = 𝒓𝒓 × 𝑚𝑚 𝒗𝒗
𝑳𝑳 = 𝑚𝑚 𝑟𝑟2𝑑𝑑𝜃𝜃𝑑𝑑𝑑𝑑
�𝒌𝒌
𝐿𝐿 = 𝑚𝑚 𝑟𝑟2𝑑𝑑𝜃𝜃𝑑𝑑𝑑𝑑
𝐿𝐿𝑚𝑚 𝑟𝑟
= 𝑟𝑟𝑑𝑑𝜃𝜃𝑑𝑑𝑑𝑑
= 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑑𝑑𝑐𝑐𝑠𝑠𝑑𝑑
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28Prof. Sergio B. MendesSummer 2018
𝐾𝐾 =12𝑚𝑚 𝑣𝑣2
𝒗𝒗 = �𝒓𝒓𝑑𝑑𝑟𝑟𝑑𝑑𝑑𝑑
+ �𝜽𝜽 𝑟𝑟𝑑𝑑𝜃𝜃𝑑𝑑𝑑𝑑
𝑣𝑣2 =𝑑𝑑𝑟𝑟𝑑𝑑𝑑𝑑
2
+ 𝑟𝑟𝑑𝑑𝜃𝜃𝑑𝑑𝑑𝑑
2=
𝑑𝑑𝑟𝑟𝑑𝑑𝑑𝑑
2
+𝐿𝐿𝑚𝑚 𝑟𝑟
2
𝐾𝐾 =12𝑚𝑚
𝑑𝑑𝑟𝑟𝑑𝑑𝑑𝑑
2
+12𝑚𝑚
𝐿𝐿𝑚𝑚 𝑟𝑟
2
![Page 29: Rotational Vectors and Angular Momentum 298 summer... · Summer 2018 Prof. Sergio B. Mendes 2 Angular Velocity actually needs to be described by a vector: • Angular velocity is](https://reader034.vdocuments.us/reader034/viewer/2022042303/5ece26e2cebd7c0f84040f8b/html5/thumbnails/29.jpg)
29Prof. Sergio B. MendesSummer 2018
𝐸𝐸 = 𝐾𝐾 + 𝑈𝑈
=12𝑚𝑚
𝑑𝑑𝑟𝑟𝑑𝑑𝑑𝑑
2
+12𝑚𝑚
𝐿𝐿𝑚𝑚 𝑟𝑟
2−𝐺𝐺 𝑀𝑀 𝑚𝑚𝑟𝑟
𝑈𝑈𝑒𝑒𝑓𝑓𝑓𝑓
𝑈𝑈𝑒𝑒𝑓𝑓𝑓𝑓
![Page 30: Rotational Vectors and Angular Momentum 298 summer... · Summer 2018 Prof. Sergio B. Mendes 2 Angular Velocity actually needs to be described by a vector: • Angular velocity is](https://reader034.vdocuments.us/reader034/viewer/2022042303/5ece26e2cebd7c0f84040f8b/html5/thumbnails/30.jpg)
30Prof. Sergio B. MendesSummer 2018
Earth’s Precession