Classical Mechanics 1 Revision

Fundamental Physical Quantities and Standard Units in Classical Mechanics

In the study of classical mechanics, identifying the correct units for physical quantities is essential for accurate measurement and calculation. The unit of speed, denoted as $v$, is defined as meters per second ($\text{m}\,\text{s}^{-1}$). For rotational motion, the unit of angular speed, represented by the Greek letter $\omega$, is expressed as radians per second ($\text{rad}\,\text{s}^{-1}$). Power is measured in Watts ($W$), where one Watt is equivalent to the transfer of one Joule per second ($\text{J}\,\text{s}^{-1}$).

Torque, represented by the symbol $\tau$, has the unit Newton-meters ($\text{N}\,\text{m}$. Linear momentum, denoted by the letter $p$, is measured in kilogram-meters per second ($\text{kg}\,\text{m}\,\text{s}^{-1}$). For rotating systems, angular momentum, represented as $\mathbf{L}$, uses the units kilogram-meters squared per second ($\text{kg}\,\text{m}^2\,\text{s}^{-1}$). These units provide the foundation for dimensional analysis and the verification of physical equations.

Kinematics and the Principles of Motion

Kinematics focuses on describing the motion of objects without considering the forces that cause that motion. A fundamental principle in kinematics is that if the velocity of an object remains constant, its acceleration must be zero. This is because acceleration is defined as the rate of change of velocity over time. In specialized cases such as free fall, the motion of an object is characterized by the fact that gravity is the only force acting upon it. This assumes the absence of other forces like air resistance.

To determine the specific behavior of an object at a precise moment, we use instantaneous quantities. If the position of an object is represented by $x$ at any time $t$, then the instantaneous velocity $v$ is mathematically defined as the first derivative of position with respect to time, or $v = \frac{dx}{dt}$. For instance, if the position of an object follows the function $x = t^2 + 4t$, the instantaneous velocity function is found by differentiating, resulting in $v = 2t + 4$. Evaluating this at a time of $t = 1.00\,\text{s}$ yields a velocity of $6.00\,\text{m}\,\text{s}^{-1}$. Similarly, if the position function is periodic, such as $x = \sin(\omega t)$ where $\omega$ is a constant, the velocity is determined as the derivative $v = \omega \cos(\omega t)$.

In one-dimensional motion with an initial velocity $v_i$ and constant acceleration $a$, the final velocity $v_f$ after a time interval $t$ can be calculated using the kinematic equation $v_f = v_i + at$. A practical application of this occurs when a body falls from rest ($v_i = 0$) with the acceleration due to gravity ($a = 9.80\,\text{m}\,\text{s}^{-2}$). After a time interval of $t = 3.00\,\text{s}$, the final velocity of the body hitting the ground is found to be $29.40\,\text{m}\,\text{s}^{-1}$.

Dynamics and Newton's Laws of Motion

Dynamics introduces the concept of forces and interactions between objects and their environment. Friction is a specific type of force that resists the motion of an object as a result of its interaction with its surroundings. The behavior of forces is governed by Newton's Laws of Motion. Newton's First Law, often called the Law of Inertia, states that in the absence of external forces, an object remains in its current state of motion; an object at rest stays at rest, and an object in motion continues moving with a constant velocity.

Newton's Second Law establishes a quantitative relationship between force and motion, stating that the net force acting on an object is directly proportional to its acceleration ($\mathbf{F} = m\mathbf{a}$). Newton's Third Law focuses on the interaction between two bodies, stating that for every action exerted by object 1 on object 2, there is a reaction force exerted by object 2 on object 1. This reaction force is always equal in magnitude and opposite in direction to the initial action force. When a system is in a state of equilibrium, the vector sum of all forces acting on the object is equal to zero.

Scalar and Vector Analysis in Classical Mechanics

Physical quantities are divided into scalars and vectors. A scalar quantity is characterized by having magnitude only, with no associated direction. In contrast, a vector quantity possesses both magnitude and direction. Analyzing vectors often involves calculating their components in space. For a vector defined as $\mathbf{r} = 3.0\hat{i} + 4.0\hat{j}$, the magnitude or length of the vector is found using the Pythagorean theorem, resulting in $r = 5.0$. The direction or angle $\theta$ of this vector is calculated using the inverse tangent of the components, resulting in $\theta = 53.13^\circ$. If a vector in the xy-plane has equal x and y components, such as $25.00\,\text{m}$, it forms an angle of $45.00^\circ$ with the positive x-axis.

Vector operations include addition and various products. When considering two vectors, such as $\mathbf{A} = 2.0\hat{i} + 2.0\hat{j}$ and $\mathbf{B} = 2.0\hat{i} - 4.0\hat{j}$, the resultant vector $\mathbf{R}$ is the sum of their components, resulting in $4.0\hat{i} - 2.0\hat{j}$. The scalar product, or dot product, of two vectors $\mathbf{A} = 3.0\hat{i} + 4.0\hat{j}$ and $\mathbf{B} = 5.0\hat{i} + 6.0\hat{j}$ is calculated as $(3.0 \times 5.0) + (4.0 \times 6.0) = 39.00$. The vector product, or cross product, of $\mathbf{B} \times \mathbf{A}$ for these same vectors results in a vector perpendicular to the plane of the original vectors, specifically $2.0\hat{k}$. These mathematical operations are descriptive of various physical phenomena: Work is defined as the scalar product of applied force and displacement vectors, while Torque and Angular Momentum are defined as the vector products of their respective constituent vectors.

Uniform Circular Motion and Projectile Dynamics

In uniform circular motion, a particle moves along a circular path of radius $r$ with a constant linear speed $v$. Even though the speed is constant, the particle experiences a centripetal acceleration ($a_c$) directed toward the center of the circle, defined by the formula $a_c = \frac{v^2}{r}$. The time it takes to complete one full revolution is the periodic time $T$, defined as $T = \frac{2\pi r}{v}$. The angular velocity $\omega$ is related to the period by the expression $\omega = \frac{2\pi}{T}$. For a particle moving with a linear speed of $v = 6.00\,\text{m}\,\text{s}^{-1}$ in a circle with a radius of $r = 3.00\,\text{m}$, the centripetal acceleration is $12.00\,\text{m}\,\text{s}^{-2}$ and the periodic time is $3.14\,\text{s}$.

Projectile motion describes objects moving in two dimensions under the influence of gravity. The horizontal range $R$ of a projectile launched with initial speed $v_i$ at an angle $\theta_i$ is given by $R = \frac{v_i^2 \sin(2\theta_i)}{g}$. The maximum height $h$ reached is given by $h = \frac{v_i^2 \sin^2(\theta_i)}{2g}$. In a scenario where $v_i = 11.00\,\text{m}\,\text{s}^{-1}$ and $\theta_i = 45.00^\circ$, the horizontal range is calculated as $12.35\,\text{m}$. If the angle is changed to $20.00^\circ$ with the same initial speed, the maximum height reached is $0.72\,\text{m}$. Crucially, in projectile motion, the horizontal component of acceleration ($a_x$) is always zero, assuming no air resistance.

Energy, Work, and Power in Systems

Energy takes multiple forms within a system. Kinetic energy is the energy of motion, while potential energy is an amount of stored energy inside the system that has the ability to transform into another kind of energy. The total energy of a system is the sum of its kinetic energy, potential energy, and internal energy. The change in gravitational potential energy ($\Delta PE_g$) when an object of mass $m$ is moved from height $y_i$ to $y_f$ is calculated by the formula $m g (y_f - y_i)$. For a mass of $m = 2.00\,\text{kg}$ moved from $y_i = 0$ to $y_f = 10.00\,\text{m}$, $\Delta PE_g$ is $196.00\,\text{J}$.

Work is calculated as the product of the applied force ($F$), the displacement ($d$), and the cosine of the angle ($\theta$) between them. The Work-Kinetic Energy Theorem states that when work is done on a system and the only change is in its speed, the net external work done on the system ($W_{ext}$) equals the change in the kinetic energy ($\Delta K.E.$). Thus, if $\Delta K.E. = 8.00\,\text{J}$, the work done is also $8.00\,\text{J}$. For elastic systems like springs, Hooke's Law states that the spring force $F_s$ is given by $F_s = -kx$, where $k$ is the spring constant and $x$ is the elongation. A spring with $k = 5000.00\,\text{N}\,\text{m}^{-1}$ elongated by $x = 0.02\,\text{m}$ experiences a force magnitude of $1.00 \times 10^2\,\text{N}$.

Power is defined as the rate at which energy is transferred or transformed. The average power ($P_{av}$) is the ratio of the change in energy ($\Delta E$) to the time interval ($\Delta t$), expressed as $P_{av} = \frac{\Delta E}{\Delta t}$. If a system undergoes an energy change of $\Delta E = 3.00\,\text{kJ}$ over a duration of $\Delta t = 10.00\,\text{s}$, the average power is calculated as $300.00\,W$. Finally, for rotating rigid bodies, the magnitude of angular momentum ($L_z$) is related to the moment of inertia ($I$) and angular speed ($\omega$) by the equation $L_z = I\omega$.