Fundamental Concepts in Mechanics: Motion, Forces, Circular Motion, and Energy

Representing Motion

Motion with Constant Acceleration (Including Free Fall)

An object with constant acceleration experiences a linearly changing velocity and a parabolic position-time graph. These principles are foundational for analyzing motion in one dimension, including free fall.

• Kinematic Equations: For constant acceleration, the following equations apply:

• $v_x = v_{x0} + a_x \Delta t$

• $x = x_0 + v_{x0} \Delta t + \frac{1}{2} a_x (\Delta t)^2$

• $v_x^2 = v_{x0}^2 + 2a_x \Delta x$

• Example: A ball dropped from rest accelerates downward at $g = 9.8\,\mathrm{m/s^2}$, following these equations.

Motion on a Ramp

An object sliding down a ramp accelerates parallel to the ramp's surface. The acceleration depends on the angle of inclination.

• $a_x = \pm g \sin \theta$

• The sign depends on the direction of the ramp and the chosen coordinate system.

• Example: A block on a $30^\circ$ incline accelerates at $a_x = g \sin 30^\circ = 4.9\,\mathrm{m/s^2}$.

Relative Motion

Velocities are relative to the observer. Relative velocity allows conversion between different frames of reference.

• For objects A and B: $v_{A/B} = v_{A/C} + v_{C/B}$

• Example: If a car moves at $15\,\mathrm{m/s}$ and a runner at $5\,\mathrm{m/s}$ in the same direction, the car's speed relative to the runner is $10\,\mathrm{m/s}$.

Forces & Newton's Laws of Motion

Definition of Force

A force is a push or pull on an object, characterized as a vector (having both magnitude and direction). Forces require an agent and can be contact or long-range (e.g., gravity).

• SI unit: newton (N)

• $1\,\mathrm{N}$ causes a $1\,\mathrm{kg}$ mass to accelerate at $1\,\mathrm{m/s^2}$.

Newton's First Law (Law of Inertia)

An object at rest remains at rest, and an object in motion continues in a straight line at constant speed unless acted upon by a net force.

• If $\vec{F} = 0$, then $\vec{a} = 0$.

• Example: A hockey puck glides on ice at constant velocity if no net force acts.

Newton's Second Law

The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass.

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

• $\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \ldots$

• Example: A $2\,\mathrm{kg}$ object with a $6\,\mathrm{N}$ net force accelerates at $3\,\mathrm{m/s^2}$.

Mass and Acceleration

Mass quantifies an object's resistance to acceleration. For two objects A and B under the same force:

• $\frac{a_A}{a_B} = \frac{m_B}{m_A}$

• Example: If object A is twice as massive as B, its acceleration is half as much under the same force.

Newton's Third Law

For every action, there is an equal and opposite reaction. Forces always occur in pairs acting on different objects.

• $\vec{F}_{A\text{ on }B} = -\vec{F}_{B\text{ on }A}$

• Example: When you push on a wall, the wall pushes back with equal force.

Objects in Contact and Free-Body Diagrams

When two objects interact, draw separate free-body diagrams for each. Action-reaction pairs are equal in magnitude and opposite in direction.

• Each object’s diagram includes all forces acting on it.

Component Form of Newton's Laws

Newton's laws are vector equations and must be written in component form for problem-solving:

• $(F_{\text{net},x}) = \sum F_x = ma_x$

• $(F_{\text{net},y}) = \sum F_y = ma_y$

• For equilibrium: $a_x = 0$, $a_y = 0$

Important Forces: Weight and Friction

• Weight: $\vec{w} = m\vec{g}$ (downward)

• Friction:

  • Static friction: $\vec{f}_s = (0 \text{ to } \mu_s n)$, direction prevents relative motion

  • Kinetic friction: $\vec{f}_k = \mu_k n$, direction opposes motion

Apparent Weight

Apparent weight is the magnitude of the contact force supporting an object (what a scale reads). It equals true weight $w = mg$ only when vertical acceleration is zero.

Terminal Speed

A falling object reaches terminal speed when the drag force balances the weight, resulting in zero acceleration:

• $\vec{a} = 0$ when $\vec{D} = \vec{w}$

Strings and Pulleys

• The tension in a rope is equal to the force pulling on it.

• In a massless rope, tension is the same at all points.

• Tension does not change over a massless, frictionless pulley.

Circular Motion, Orbits & Gravity

Uniform Circular Motion

An object moving in a circle at constant speed experiences a changing direction of velocity. The acceleration (centripetal) is directed toward the center:

• $a = \frac{v^2}{r}$

• Example: A car turning in a circle of radius $10\,\mathrm{m}$ at $5\,\mathrm{m/s}$ has $a = 2.5\,\mathrm{m/s^2}$ toward the center.

Universal Gravitation

Any two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of their separation:

• $F_{12} = F_{21} = \frac{G m_1 m_2}{r^2}$

• $G = 6.67 \times 10^{-11}\,\mathrm{N\cdot m^2/kg^2}$

Planetary Gravity

The gravitational force between a planet and a mass on its surface is:

• $F_{\text{planet on m}} = \frac{GM_{\text{planet}} m}{R_{\text{planet}}^2}$

• Free-fall acceleration: $g_{\text{planet}} = \frac{GM_{\text{planet}}}{R_{\text{planet}}^2}$

Apparent Weight and Weightlessness

In circular motion, the net force must point to the center. Apparent weight is not always equal to true weight, especially in non-inertial frames.

Orbital Motion

A satellite in a circular orbit of radius $r$ around a mass $M$ moves at speed:

• $v = \sqrt{\frac{GM}{r}}$

• Period-radius relation: $T^2 = \left(\frac{4\pi^2}{GM}\right) r^3$

• Both equations are independent of satellite mass.

Describing Circular Motion

• Angular displacement: $\Delta \theta = \theta_f - \theta_i$

• Angular velocity: $\omega = \frac{\Delta \theta}{\Delta t}$

• Angular acceleration: $\alpha = \frac{\Delta \omega}{\Delta t}$

• 1 revolution = $2\pi$ radians

• $\omega = \frac{2\pi}{T} = 2\pi f$

Newton's Second Law for Rotational Motion

If a net torque $\tau_{\text{net}}$ acts on an object, it experiences angular acceleration:

• $\alpha = \frac{\tau_{\text{net}}}{I}$

• Linear and angular speeds: $v = \omega r$

Torque

Torque causes angular acceleration:

• $\tau = rF \sin \phi$

• Maximum when force is perpendicular to the lever arm.

Moment of Inertia

The moment of inertia $I$ quantifies rotational inertia:

• $I = \sum m_i r_i^2$

• Depends on mass distribution and axis of rotation.

• Units: $\mathrm{kg\cdot m^2}$

Center of Gravity

The center of gravity is the point where gravity acts on an object. Its position is:

• $x_{cg} = \frac{x_1 m_1 + x_2 m_2 + x_3 m_3 + \ldots}{m_1 + m_2 + m_3 + \ldots}$

Rotation about a Fixed Axis

When a net torque is applied to an object about a fixed axis:

• $\alpha = \frac{\tau_{\text{net}}}{I}$

Static Equilibrium

An object in static equilibrium has no net force and no net torque:

• $\sum F_x = 0$

• $\sum F_y = 0$

• $\sum \tau = 0$

• The pivot point for torque can be chosen for convenience.

Springs and Hooke's Law

A spring exerts a force proportional to its displacement from equilibrium:

• Hooke's Law: $F_{\text{sp}} = -k \Delta x$

• $k$ is the spring constant (stiffer springs have larger $k$).

Momentum

Linear Momentum and Impulse

• Momentum: $\vec{p} = m\vec{v}$

• Impulse: $J = \text{area under force-time curve}$

• $\Delta p = J$

• Angular momentum: $L = I\omega$ (rotational analog of $p = mv$)

Energy & Work

Work

Work is the transfer of energy via mechanical forces:

• $W = Fd = Fd \cos \theta$

• Only the component of force parallel to displacement does work.

Kinetic and Potential Energy

• Kinetic energy: $K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$ (translational + rotational)

• Potential energy:

  • Gravitational: $U_g = mgy$

  • Elastic: $U_s = \frac{1}{2}kx^2$

• Thermal energy: Manifested as heat.

Conservation Laws

Conservation of energy and momentum are fundamental principles in physics, ensuring that total energy and momentum remain constant in isolated systems.