Physics 321
Hour 2
Applying Newton’s Laws of Motion
Manipulating the 2nd Law
It is often better to turn one 2nd order equation
Into two 1st order equations:
𝐹 = 𝑚𝑥
𝐹 = 𝑚𝑣,
𝑣=𝑥
or
𝐹 = 𝑝,
𝑝 = 𝑚𝑥
Manipulating the 2nd Law
We can also express F in terms of spatial
derivatives:
𝑑𝑣 𝑑𝑥
𝑑𝑣 1 𝑑𝑝
𝐹 = 𝑚𝑣 = 𝑚
=𝑝
= 𝑝
𝑑𝑥 𝑑𝑡
𝑑𝑥 𝑚 𝑑𝑥
1 𝑑𝑝2 𝑑𝑇
𝐹=
=
2𝑚 𝑑𝑥
𝑑𝑥
A Numerical Strategy
We use a new variable, u, to parameterize the motion:
𝐹=𝑝
𝑝 = 𝑚𝑥
∆𝑡
∆𝑥 = 𝑝 ≡ 𝑝∆𝑢
𝑚
∆𝑝 = 𝐹∆𝑡 = 𝐹𝑚∆𝑢
So:
∆𝑡 = 𝑚∆𝑢
∆𝑥 = 𝑝∆𝑢
∆𝑝 = 𝐹𝑚∆𝑢
The Relativistic Version
We know E=mc2. What if mass increases when
kinetic energy increases?
∆𝐾 𝐹∆𝑥 𝐹𝑝
∆𝑚 = 2 = 2 = 2 ∆𝑢
𝑐
𝑐
𝑐
So:
∆𝑡 = 𝑚∆𝑢
∆𝑥 = 𝑝∆𝑢
∆𝑝 = 𝐹𝑚∆𝑢
𝐹𝑝
∆𝑚 = 2 ∆𝑢
𝑐
The Physics 121 Strategy
• Draw a free body diagram for each part of
the system
• Sum forces: 𝐹 = 𝑚𝑎
• Sum torques: 𝜏 = 𝐼 𝛼
• Relate forces to torques: 𝜏 = 𝑟 × 𝐹