College Physics I: Newton’s Laws, Circular Motion, Work & Energy, and Momentum – Study Notes

Ch5 Application of Newton’s Law

Newton’s Laws of Motion

Newton’s laws describe the relationship between the motion of an object and the forces acting on it. These laws are foundational to classical mechanics.

• Newton’s First Law (Law of Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.

• Newton’s Second Law: The net force on an object is equal to the mass of the object multiplied by its acceleration. $\vec{F}_{\text{net}} = m\vec{a}$ $F_{\text{net},x} = \sum F_{i,x} = m a_x$ $F_{\text{net},y} = \sum F_{i,y} = m a_y$

• Newton’s Third Law: For every action, there is an equal and opposite reaction. $\vec{F}_{A\text{ on }B} = -\vec{F}_{B\text{ on }A}$

Friction Forces

• Maximum static friction: The maximum force before sliding begins. $f_{s,\text{max}} = \mu_s n$

• Kinetic friction: The force opposing motion once sliding has started. $f_k = \mu_k n$

Ch6 Circular Motion and Gravitation

Uniform Circular Motion

Objects moving in a circle at constant speed experience a net force directed toward the center of the circle (centripetal force).

• Centripetal force: $F_{\text{net}} = m \frac{v^2}{R}$

Newton’s Law of Gravitation

Every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Gravitational force: $F_g = G \frac{m_1 m_2}{r^2}$ $G = 6.674 \times 10^{-11} \ \text{N} \cdot \text{m}^2/\text{kg}^2$

• Gravitational acceleration at Earth’s surface: $g = G \frac{m_E}{R_E^2}$ where $m_E$ is Earth’s mass and $R_E$ is Earth’s radius.

Satellite Orbits

• Satellite speed: $v = \sqrt{\frac{GM}{r}}$

• Satellite period: $T = \frac{2\pi r}{v} = \frac{2\pi r^{3/2}}{\sqrt{GM}}$

Ch7 Work and Energy

Kinetic and Potential Energy

Energy is the capacity to do work. Mechanical energy includes kinetic and potential forms.

• Kinetic energy: $K = \frac{1}{2} m v^2$

• Potential energy:

  • Gravitational: $U_g = mgy$

  • Elastic (spring): $U_{el} = \frac{1}{2} k x^2$

Work and the Work-Energy Theorem

• Work: $W = F_{\parallel} s$

• Work-energy theorem: $W_{\text{total}} = K_f - K_i = \Delta K$

Total Mechanical Energy and Conservation

• Total mechanical energy: $E = K + U_g + U_{el} = \frac{1}{2} m v^2 + mgy + \frac{1}{2} k x^2$

• Conservation (only conservative forces): $E_f = E_i$ or $K_f + U_{g,f} + U_{el,f} = K_i + U_{g,i} + U_{el,i}$

• With non-conservative forces: $W_{\text{other}} = E_f - E_i$ or $W_{\text{other}} = (K_f + U_{g,f} + U_{el,f}) - (K_i + U_{g,i} + U_{el,i})$

Ch8 Momentum (up to 8.4)

Momentum and Its Conservation

Momentum is a vector quantity defined as the product of mass and velocity. In the absence of external forces, the total momentum of a system remains constant.

• Momentum: $\vec{P} = m \vec{v}$ (Unit: kg·m/s)

• Vector form: $\vec{P} = (P_x, P_y) = (m v_x, m v_y)$

• Total momentum of a system: $\vec{P}_{\text{tot}} = \vec{P}_A + \vec{P}_B + \vec{P}_C + \ldots$

• Conservation of momentum: $\vec{P}_{f,\text{tot}} = \vec{P}_{i,\text{tot}}$ (if net external $F_{\text{net}} = 0$)

Collisions

• Elastic collisions:

  • Momentum conservation: $m_A v_{A,i} + m_B v_{B,i} = m_A v_{A,f} + m_B v_{B,f}$

  • Kinetic energy conservation: $\frac{1}{2} m_A v_{A,i}^2 + \frac{1}{2} m_B v_{B,i}^2 = \frac{1}{2} m_A v_{A,f}^2 + \frac{1}{2} m_B v_{B,f}^2$

  • Relative velocity reverses sign: $v_{B,f} - v_{A,f} = - (v_{B,i} - v_{A,i})$

  • Special cases:

    • If $v_{B,i} = 0$: $v_{A,f} = \frac{m_A - m_B}{m_A + m_B} v_{A,i}$, $v_{B,f} = \frac{2 m_A}{m_A + m_B} v_{A,i}$

• Inelastic collisions: Only momentum is conserved, not kinetic energy. $m_A v_{A,i} + m_B v_{B,i} = m_A v_{A,f} + m_B v_{B,f}$

• Completely inelastic collision: Objects stick together after collision.

  • If $v_{B,i} = 0$: $v_f = \frac{m_A}{m_A + m_B} v_{A,i}$

  • Fraction of kinetic energy remaining: $\frac{K_f}{K_i} = \frac{m_A}{m_A + m_B}$