Dynamics in 1D and Circular Motion: Study Notes

Dynamics in 1D (Ch 5–7)

Newton's Laws and Useful Equations

Newton's laws of motion form the foundation for analyzing the dynamics of objects in one dimension. Understanding these laws and their mathematical representations is essential for solving physics problems related to force and motion.

• Newton's Second Law: The net force on an object is equal to its mass times its acceleration. Equation: $F_{net} = ma$ In component form: $F_{net,x} = ma_x$, $F_{net,y} = ma_y$

• Force of Gravity: The gravitational force acting downward on an object. Equation: $F_g = mg$ (downward)

• Static Friction Force: The force that prevents motion up to a maximum value. Equation: $f_s \leq \mu_s F_N$ (direction as necessary to prevent motion) Important: $f_s$ is not always equal to $\mu_s F_N$; it can be less.

• Kinetic Friction Force: The force opposing motion between surfaces in relative motion. Equation: $f_k = \mu_k F_N$ (direction opposite the motion)

• Kinematic Equations for Constant Acceleration:

  • $v = v_0 + a t$

  • $x = x_0 + v_0 t + \frac{1}{2} a t^2$

  • $v^2 = v_0^2 + 2a(x - x_0)$

Example: Calculating the acceleration of an elevator using a free-body diagram and Newton's second law.

Problem-Solving Strategy

Drawing a free-body diagram (FBD) is the first and most important step in solving Newton's laws problems. It helps visualize all forces acting on an object and their directions.

• Identify all forces acting on the object.

• Draw the FBD, labeling each force.

• Apply Newton's second law in each direction.

• Solve for the unknowns.

Vector Components

Vectors are quantities with both magnitude and direction. Decomposing vectors into components simplifies problem-solving in physics.

• The student will be able to decompose a given vector into its x- and y-components and write the component vectors using unit vectors. Equation: $\vec{A} = A_x \hat{i} + A_y \hat{j}$

• The student will be able to determine the magnitude and direction of a vector from its components. Equation: $A = \sqrt{A_x^2 + A_y^2}$ Direction: $\theta = \tan^{-1}(A_y / A_x)$

Example: Finding the components and magnitude of a displacement vector.

Newton's Third Law

Newton's third law states that for every action, there is an equal and opposite reaction. Action and reaction forces act on different objects.

• Action and reaction forces are equal in magnitude and opposite in direction.

• For example, a table exerts a normal force upward on a resting object, while the object exerts a downward force on the table.

• Acceleration depends on both force and mass.

Key Concepts in Dynamics

Equilibrium and Net Force

An object is in equilibrium if the net force acting on it is zero. This means the object is either at rest or moving with constant velocity.

• Equilibrium: $F_{net} = 0$

• Forces in equilibrium: $\sum F_x = 0$, $\sum F_y = 0$

Types of Forces

Understanding different types of forces is crucial for analyzing physical situations.

• Gravity: $F_g = mg$

• Normal Force: Perpendicular contact force from a surface.

• Tension: Force transmitted through a string or rope.

• Friction: Opposes relative motion between surfaces.

Units of Force

The SI unit of force is the Newton (N).

• $1 \text{ N} = 1 \text{ kg} \cdot \text{m} / \text{s}^2$

Uniform Circular Motion

Forces in Circular Motion

When an object moves in a circle at constant speed, it experiences a centripetal force directed toward the center of the circle.

• Centripetal Force: $F_{c} = m a_{c}$, where $a_{c} = \frac{v^2}{r}$

• Direction: Always points toward the center of the circle.

• Not a new type of force: Centripetal force is provided by existing forces (gravity, tension, friction).

Example: Rounding corners on highways, loop-the-loop in amusement parks, circular orbits.

Work and Energy

Work Done by a Force

Work is done when a force moves an object over a distance. The work done by a constant force is given by the dot product of force and displacement.

• Work by a constant force: Equation: $W = \vec{F} \cdot \Delta \vec{x}$

• Dot Product: $\vec{A} \cdot \vec{B} = AB \cos \theta$

• In component form: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

• Work can be positive, negative, or zero depending on the angle between force and displacement.

Work-Energy Principle

The work done on an object is equal to the change in its kinetic energy.

• Equation: $W = \Delta KE$

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

Application: Use the work-kinetic energy relationship to calculate initial or final velocities of an object.

Summary Table: Types of Forces

Type of Force	Symbol	Direction	Equation
Gravity	$F_g$	Downward	$F_g = mg$
Normal Force	$F_N$	Perpendicular to surface	Depends on situation
Tension	$T$	Along string/rope	Depends on forces acting
Friction (Static)	$f_s$	Opposes motion	$f_s \leq \mu_s F_N$
Friction (Kinetic)	$f_k$	Opposes motion	$f_k = \mu_k F_N$
Centripetal Force	$F_c$	Toward center	$F_c = m \frac{v^2}{r}$

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples and applications have been added to illustrate concepts.