Chapter 1-5 Review: Momentum, Work, Energy, and Center of Mass

Linear and Angular Momentum

Momentum can be linear or angular; these are the two distinct types of momentum discussed.
Linear momentum, typically described as
p = m v
Angular momentum about an axis is described as
\mathbf{L} = I \boldsymbol{\omega}
Torque about a point or axis is defined as
\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}
or, for magnitude, \tau = r F \sin\theta
The distance between the line of action of the force and the rotation axis matters; this lever arm determines the torque.
When the force acts through the center of mass (CM), the torque about the CM is zero, so there is no angular acceleration from that force about the CM. This is illustrated by saying momentum due to linear motion is rarely through the center of mass and may not produce torque about the CM.
Practical intuition: pushing on a door hinge may break the hinges, but if you push through the hinge line, rotation about the hinge can be minimal or zero depending on geometry; what matters is the lever arm relative to the axis.
A person on ice can start spinning when they generate initial angular momentum; once spinning, angular momentum tends to be conserved in the absence of external torques.
To start spinning, muscular forces must act to create initial angular momentum; once spinning, the center of mass frame and ground interaction become important for how the motion continues.
After spinning starts, the person’s foot can be under the base of support; the ground reaction force can pass roughly through the CM, reducing external torque about the CM.
Gravity acts on the CM (downward) and contributes forces, but these forces do not necessarily create torque about the CM unless they cause a lever arm relative to the CM in a rotating frame.
Overall, the angular motion and linear motion are linked via forces, torques, and the distribution of mass (moment of inertia). The onset of rotation involves muscular work and energy input, whereas sustained rotation depends on angular momentum conservation in the presence of external torques.

Work and Energy: Core Concepts

Work and energy are the two central concepts for an energy-based perspective.
Work is defined as the transfer of energy via a force causing displacement:
W = \mathbf{F} \cdot d
For constant force over displacement, this reduces to
W = F \; d
Example: applying a force of 100\ \text{N} over a displacement of 10\ \text{m} yields
W = 100 \times 10 = 1000\ \text{J} (joules).
The system doing work could be a human body, a muscle, or a set of bodies lifting an object (e.g., lifting a couch).
Energy conversion is not free; there is an energetic cost to produce work due to metabolism. The chemical energy from the food we eat is converted through metabolic processes into work done by muscles.
Efficiency is not 100%: the conversion from chemical energy to mechanical work is imperfect, so some energy is lost as heat or tapers into other non-useful forms.
There are two main sources of potential energy in this discussion; the primary one is gravity, referred to as gravitational potential energy. In physics, this is typically called gravitational potential energy,
U = m g h
where $m$ is mass, $g$ is acceleration due to gravity, and $h$ is height.
The term “potential energy” is used to describe energy stored due to position in a force field; gravity is the common example here.
Gravity converts potential energy to kinetic energy as an object moves downward:
- Gravity acts on the CM; as the object descends, potential energy decreases and kinetic energy typically increases.
- The velocity under gravity from rest after time t is roughly
  v \approx g t
  with g \approx 9.81\ \text{m s}^{-2}
- After 1 second of free fall from rest, the speed is about 9.81\ \text{m s}^{-1}; after 2 seconds, about 19.62\ \text{m s}^{-1}, neglecting air resistance.
When you convert potential energy to kinetic energy (and vice versa), you are exchanging energy between position and motion of the CM.
In a bouncing or rhythmic motion (e.g., swinging), the velocity of the center of mass oscillates, and the height (potential energy) changes accordingly.
The ramp metaphor: instead of a stereotypical ascent, imagine a ramp that helps convert muscular energy into moving to a higher height; you start at a low height, perform muscular actions to climb to a higher height, and gravity then acts to convert that potential energy back to kinetic energy as you descend.
Energetically, you transition from standing still at a certain height to a configuration with higher potential energy, using muscle work to increase height; then gravity converts that potential energy back to kinetic energy as you move.
As with any real system, there are intermediate steps and energy transfers that are not perfectly efficient, including losses to heat, internal work, and non-conservative forces.

Interplay and Real-World Implications

The interplay between linear momentum, angular momentum, torque, work, and energy explains motions from athletic spins to walking, running, and jumping.
Understanding torque and lever arms helps explain why forces that do not act through the CM can produce rotation, whereas forces through the CM primarily affect translation.
Conservation of angular momentum explains why a figure skater can spin faster by pulling their limbs in (reducing the moment of inertia I while keeping L constant):
Li = Ii \omegai = Lf = If \omegaf with If < Ii \Rightarrow \omegaf > \omegai
Ground reaction forces can act through the CM to minimize external torque, affecting how rotation evolves without adding additional angular impulse.
The metabolic cost of movement implies that our bodies are optimized to convert chemical energy into mechanical work efficiently, but real-world efficiency is limited; some energy is inevitably lost as heat.
Practical relevance includes athletic training, biomechanics, and design of machines where energy efficiency and force application are critical.

Quick Reference Formulas (LaTeX)

Linear momentum: p = m v
Angular momentum: \mathbf{L} = I \boldsymbol{\omega}
Torque: \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}\quad\text{or}\quad \tau = r F \sin\theta
Work: W = \mathbf{F} \cdot d = F d
Gravitational potential energy: U = m g h
Kinetic energy: K = \tfrac{1}{2} m v^2
Total mechanical energy (no non-conservative losses): E = K + U
Free-fall velocity approximation: v \approx g t\quad(g \approx 9.81\ \text{m s}^{-2})
Conservation of angular momentum (spinning): Li = Lf \Rightarrow Ii \omegai = If \omegaf

Connections to Prior Concepts and Real-World Relevance

This material connects linear and rotational dynamics, the role of force direction relative to the CM, and energy methods in biomechanics and sports.
Practical applications include athletic technique optimization, understanding energy transfer during movements, and designing systems that maximize useful work while minimizing energy loss.
Philosophical aspect: energy is conserved in idealized systems; real systems show irreversibility and dissipation, highlighting limits of ideal models in describing living organisms and complex motions.