Physics Study Guide: Dynamics, Work, Energy, and Momentum (Chapters 8–11)

Dynamics II: Motion in a Plane

Circular Motion and Newton's Second Law

Circular motion involves objects moving along a curved path, typically a circle. Newton's Second Law applies to such motion, especially when analyzing forces directed toward the center (centripetal forces).

• Centripetal Acceleration: For an object moving in a circle of radius r with speed v, the acceleration toward the center is given by $a_c = \frac{v^2}{r}$.

• Newton's Second Law in Circular Motion: If the +x-axis points to the center, then $a_x = +a_{centripetal}$ and $\sum F_x = m a_x = m a_c$.

• Example: A car turning in a circle experiences a net force toward the center, provided by friction between tires and road.

Work and Kinetic Energy

Work from Individual and Multiple Forces

Work is the energy transferred by a force acting over a distance. It can be calculated for constant or variable forces, and summed for multiple forces.

• Dot Product for Work: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z = AB \cos \theta$

• Work by a Constant Force: $W = F \Delta r \cos \theta$

• Work by a Variable Force: $W = \int \vec{F} \cdot d\vec{r}$

• Work by Kinetic Friction: $W_{fk} = -f_k \Delta r$

• Displacement Vector: $d\vec{r} = dx \hat{i} + dy \hat{j} + dz \hat{k}$

• Example: Lifting a box vertically involves work against gravity: $W = m g h$.

Work-Kinetic Energy Theorem

The net work done on an object equals the change in its kinetic energy.

• Theorem: $W_{net} = \Delta K$

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

• Example: If a force accelerates a mass from rest, the work done equals its final kinetic energy.

Energy and Conservation

Potential and Kinetic Energy

Energy can be stored as kinetic, gravitational potential, or elastic potential energy. Conservation of energy relates these forms and accounts for work done by non-conservative forces.

• Gravitational Potential Energy: $U_g = m g y$

• Elastic Potential Energy (Spring): $U_{sp} = \frac{1}{2} k (\Delta s)^2$

• Power: $\text{Power} = \frac{dE}{dt}$

• Constant Power: $\text{Power} = \frac{\Delta E}{\Delta t} = \frac{W}{\Delta t}$

• Energy Conservation Equation: $\sum W_{ext} + \sum W_{NC} = \Delta K + \Delta U_g + \Delta U_{sp}$

• Expanded Form: $\sum W_{ext} + \sum W_{NC} = \left( \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 \right) + (m g y_f - m g y_i) + \left( \frac{1}{2} k (\Delta x_f)^2 - \frac{1}{2} k (\Delta x_i)^2 \right)$

• Change in Energy: $\Delta U = U_f - U_i$, $\Delta K = K_f - K_i$

• Example: A block sliding down a hill converts potential energy to kinetic energy, minus any work done by friction.

Forces and Their Magnitudes

Types of Forces

Several forces are commonly encountered in physics problems, each with characteristic equations.

• Maximum Static Friction: $f_{s, max} = \mu_s N$

• Kinetic Friction: $f_k = \mu_k N$

• Gravity: $F_G = m g$

• Spring Force: $F_{sp} = k |\Delta s|$

• Example: A block on an incline experiences gravity, friction, and possibly a spring force.

Momentum and Impulse

Momentum Conservation and Impulse

Momentum is a fundamental quantity conserved in isolated systems. Impulse relates the change in momentum to the force applied over time.

• Momentum: $\vec{p} = m \vec{v}$

• Total Momentum: $\vec{P} = m_1 \vec{v}_1 + m_2 \vec{v}_2 + \cdots$

• Impulse: $\Delta \vec{p} = \int_{t_i}^{t_f} \vec{F} dt = \vec{F}_{ave} \Delta t$

• Change in Total Momentum: $\Delta \vec{P} = \int_{t_i}^{t_f} \vec{F}_{External} dt$

• Conservation in Isolated Systems: If $\vec{F}_{external} = 0$, then $\Delta \vec{P} = 0$ and $\vec{P}_i = \vec{P}_f$

• Example: Two colliding carts on a frictionless track conserve total momentum.

Momentum Conservation in 1-D and 2-D

Momentum conservation applies in both one and two dimensions, requiring vector addition of momentum components.

• Conservation Equations:

  • $m_1 v_{1x,i} + m_2 v_{2x,i} + \cdots = m_1 v_{1x,f} + m_2 v_{2x,f} + \cdots$

  • $m_1 v_{1y,i} + m_2 v_{2y,i} + \cdots = m_1 v_{1y,f} + m_2 v_{2y,f} + \cdots$

• Example: A two-dimensional collision (e.g., billiard balls) requires conservation in both x and y directions.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Work (constant force)	$W = F \Delta r \cos \theta$	Work done by a constant force
Work (variable force)	$W = \int \vec{F} \cdot d\vec{r}$	Work done by a force that varies with position
Kinetic Energy	$K = \frac{1}{2} m v^2$	Energy of motion
Gravitational Potential Energy	$U_g = m g y$	Energy due to height in a gravitational field
Elastic Potential Energy	$U_{sp} = \frac{1}{2} k (\Delta s)^2$	Energy stored in a spring
Momentum	$\vec{p} = m \vec{v}$	Product of mass and velocity
Impulse	$\Delta \vec{p} = \int \vec{F} dt$	Change in momentum due to force over time
Conservation of Momentum	$\vec{P}_i = \vec{P}_f$	Momentum is conserved in isolated systems
Friction (static/kinetic)	$f_{s, max} = \mu_s N$, $f_k = \mu_k N$	Maximum static and kinetic friction forces
Spring Force	$F_{sp} = k |\Delta s|$	Force exerted by a spring

Additional info: Academic context and examples were added to clarify brief points and equations, ensuring the notes are self-contained and suitable for exam preparation.