Physics 321
Hour 3
Application of Newton’s Laws/Drag Forces
Consequences of Relativity
• Momentum becomes
𝑝 = 𝑚 𝑣 = 𝑚0 𝛾 𝑣
• Kinetic energy becomes
𝑇 = 𝑚𝑐 2 − 𝑚0 𝑐 2
• Potential energy becomes hazy
• Momentum is conserved “asymptotically”
Problem 1
A mass hangs from a string from the
roof of a train car accelerating with 𝑎.
What is θ?
θ
𝑎
Problem 2
A sphere of mass m, radius R, and
moment of inertia I rolls on a horizontal
surface without slipping when a tension
T is applied. What is the acceleration?
𝑇
Problem 3
A large mass M and a small mass m are
𝑇
stacked on each other. A tension T is
applied to the large mass. The coefficient
of static friction is μs and the coefficient of sliding friction is
μ. How big can T become without the top mass slipping?
Drag Forces
1) Forces of the form 𝐹 = −|𝑓 … |𝑣 cause an object to
slow down. Why?
Drag Forces
2) A typical drag force can have linear and quadratic
components:
How can we write this in one dimension (eg, for free
fall)?
𝐹𝑑 = −𝑐1 𝑦 + 𝑐2 𝑦 2
How can we write this in three dimensions?
𝐹𝑑 = −𝑐1 𝑣 − 𝑐2 𝑣 𝑣
Free fall with linear drag
3) The differential equation:
𝑚𝑣 = −𝑚𝑔 − 𝑐1 𝑣
𝑚𝑔
4) Terminal velocity – when 𝑣 = 0, 𝑣 = −
𝑐1
Mathematica
1. Drag3-1.nb
2. LinvsQuad.nb