Dynamics in One Dimension and Work-Energy Principles

Dynamics in 1D (Ch 5–7)

Newton's Laws and Problem-Solving Strategies

Dynamics in one dimension involves understanding how forces affect the motion of objects along a straight line. Newton's laws of motion form the foundation for analyzing such problems.

• Newton’s Second Law: The net force on an object is equal to the mass times its acceleration.

$\vec{F}_{\text{net}} = m\vec{a}$

• In component form: $F_{\text{net},x} = m a_x$, $F_{\text{net},y} = m a_y$

• Force of gravity: $\vec{F}_g = m\vec{g}$ (downward)

• Static friction force: $f_s \leq \mu_s F_N$ (maximum value of $f_s$ is $\mu_s F_N$; direction as necessary to prevent motion)

• Kinetic friction force: $f_k = \mu_k F_N$ (direction opposite the motion)

• Kinematic equations for constant acceleration:

$\begin{align*} v &= v_0 + a t \\ x &= x_0 + v_0 t + \frac{1}{2} a t^2 \\ v^2 &= v_0^2 + 2a(x - x_0) \end{align*}$

• Acceleration, $a$: Links force to kinematics. Find $a$ first, then use kinematics to find $d$, $t$, $v$, etc.

• Equilibrium: An object is in equilibrium if $\sum \vec{F} = 0$ (at rest or moving with constant velocity).

Problem-Solving Strategy:

1. Draw a free-body diagram (FBD) for the object.

2. Apply Newton’s second law in the $x$ and $y$ directions.

3. Solve for the unknowns.

Note: $f_s$ is not always equal to $\mu_s F_N$; it is only at its maximum value when the object is on the verge of moving.

Vector Components and Newton’s Third Law

Vectors are essential in physics for representing quantities with both magnitude and direction. Newton’s third law describes the interaction between pairs of objects.

• Decomposing Vectors: Any vector $\vec{A}$ can be written in terms of its $x$ and $y$ components:

$\vec{A} = A_x \hat{i} + A_y \hat{j}$

• Magnitude of a vector: $A = \sqrt{A_x^2 + A_y^2}$

• Newton’s Third Law: For every action, there is an equal and opposite reaction. Action and reaction forces act on different objects.

Example: A book resting on a table exerts a downward force (weight) on the table, and the table exerts an equal upward normal force on the book.

Key Concepts in Dynamics

• Equilibrium: Net force is zero; object is at rest or moves with constant velocity.

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

• Newton’s Second Law: Net force causes acceleration.

• Newton’s Third Law: Forces always occur in pairs, equal in magnitude and opposite in direction.

• Types of Forces: Gravity, spring force, tension, normal force, friction.

• Free-Body Diagrams: Essential for visualizing all forces acting on an object.

• Units of Force: Newton ($N$), where $1\ N = 1\ kg \cdot m/s^2$

Uniform Circular Motion

Forces and Acceleration 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 acceleration: $a_c = \frac{v^2}{r}$, always points toward the center.

• Centripetal force: $F_{c} = m a_c = m \frac{v^2}{r}$

• Any force (gravity, tension, friction) can provide the required centripetal force.

Applications:

• Rounding corners on highways

• Loop-the-loop (e.g., Ferris wheel)

• Circular orbits

Note: Centripetal force is not a new type of force; it is the name for the net force causing circular motion.

Work and Energy

Work Done by a Force

Work is the transfer of energy by a force acting over a distance. It is a scalar quantity.

• Work by a constant force:

$W = \vec{F} \cdot \Delta \vec{x}$

• For a variable force:

$W = \int_{x_i}^{x_f} F_x dx$

• 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 $\vec{F}$ and $\Delta \vec{x}$.

Work-Energy Principle

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

$W = \Delta KE$

$KE = \frac{1}{2} m v^2$

Application: The work-kinetic energy relationship can be used to calculate the initial or final velocity of an object when the net work done is known.