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1Prof. Sergio B. MendesSummer 2018
Chapter 11 of Essential University Physics, Richard Wolfson, 3rd Edition
Rotational Vectors and Angular Momentum
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
3Prof. Sergio B. MendesSummer 2018
Angular Acceleration is also a vector: ��𝛼 ≡ 𝑑𝑑𝜔𝜔
𝑑𝑑𝑑𝑑
4Prof. Sergio B. MendesSummer 2018
A Different Kind of Product Between Two Vectors
A Math Break:
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
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
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
8Prof. Sergio B. MendesSummer 2018
Back to the Physics:
9Prof. Sergio B. MendesSummer 2018
Torque is the cross product of the two vectors: 𝒓𝒓 and 𝑭𝑭
𝝉𝝉 ≡ 𝒓𝒓 × 𝑭𝑭
𝒓𝒓
𝑭𝑭𝜃𝜃
𝝉𝝉
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
11Prof. Sergio B. MendesSummer 2018
Definition of Angular Momentum:
𝑳𝑳 ≡ 𝒓𝒓 × 𝒑𝒑
𝒓𝒓
𝒑𝒑𝜃𝜃
𝑳𝑳
𝑳𝑳 ⊥ 𝒓𝒓
𝑳𝑳 ⊥ 𝒑𝒑
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
13Prof. Sergio B. MendesSummer 2018
In linear motion we had:
𝒑𝒑 = 𝑚𝑚 𝒗𝒗
now, in rotational motion we have:
𝑳𝑳 = 𝐼𝐼 𝝎𝝎
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
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
16Prof. Sergio B. MendesSummer 2018
In linear motion we had
𝑭𝑭𝒏𝒏𝒏𝒏𝒏𝒏,𝒏𝒏𝒆𝒆𝒏𝒏 =𝑑𝑑𝑷𝑷𝑑𝑑𝑑𝑑
now, in rotational motion we have:
𝝉𝝉𝒏𝒏𝒏𝒏𝒏𝒏,𝒏𝒏𝒆𝒆𝒏𝒏 =𝑑𝑑𝑳𝑳𝑑𝑑𝑑𝑑
17Prof. Sergio B. MendesSummer 2018
𝑳𝑳 = 𝐼𝐼 𝝎𝝎 = 𝒄𝒄𝒄𝒄𝒏𝒏𝒄𝒄𝒏𝒏𝒂𝒂𝒏𝒏𝒏𝒏𝝉𝝉𝒏𝒏𝒏𝒏𝒏𝒏,𝒏𝒏𝒆𝒆𝒏𝒏 = 0
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
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
21Prof. Sergio B. MendesSummer 2018
𝑳𝑳 = 𝐼𝐼 𝝎𝝎 = 𝒄𝒄𝒄𝒄𝒏𝒏𝒄𝒄𝒏𝒏𝒂𝒂𝒏𝒏𝒏𝒏𝝉𝝉𝒏𝒏𝒏𝒏𝒏𝒏,𝒏𝒏𝒆𝒆𝒏𝒏 = 0
23Prof. Sergio B. MendesSummer 2018
Gyroscopes
25Prof. Sergio B. MendesSummer 2018
Gyroscope Precession
26Prof. Sergio B. MendesSummer 2018
Addendum to Chapter 8 about Gravitation
𝑭𝑭 = − 𝐺𝐺𝑀𝑀 𝑚𝑚𝑟𝑟2
��𝑟
𝝉𝝉 = 𝒓𝒓 × 𝑭𝑭 = 0
𝑳𝑳 = 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑑𝑑𝑐𝑐𝑠𝑠𝑑𝑑 = 𝒓𝒓 × 𝑚𝑚 𝒗𝒗
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
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
29Prof. Sergio B. MendesSummer 2018
𝐸𝐸 = 𝐾𝐾 + 𝑈𝑈
=12𝑚𝑚
𝑑𝑑𝑟𝑟𝑑𝑑𝑑𝑑
2
+12𝑚𝑚
𝐿𝐿𝑚𝑚 𝑟𝑟
2−𝐺𝐺 𝑀𝑀 𝑚𝑚𝑟𝑟
𝑈𝑈𝑒𝑒𝑓𝑓𝑓𝑓
𝑈𝑈𝑒𝑒𝑓𝑓𝑓𝑓
30Prof. Sergio B. MendesSummer 2018
Earth’s Precession
