Physics Lecture Notes: Momentum, Energy, and Motion

Newton's First Law

• Inertia: Describes the difficulty in changing an object's motion. It is caused by mass.

• Newton's First Law: Specifies that an object in motion tends to stay in motion, and an object at rest tends to stay at rest, unless acted upon by an outside force.

• Mass: A physical property that relates to inertia ($mass = inertia$). It is a property of the matter in an object and does not change based on location or position in the universe.

• Weight: A force related to gravity. It is the experience of Earth's gravitational pull.

• Force ($F$ or $\vec{F}$): A push or pull that causes an object to change its motion. It is a vector.

• Vector: A quantity that contains both magnitude (strength) and direction. In diagrams, the length of the arrow represents strength.

• Units: The Newton ($N$) is the unit of force, where $1\,N = 1\,kg \cdot m/s^2$.

• Mechanical Equilibrium: A condition where an object experiences no change in its motion. This occurs when the sum of all forces acting on the object is zero ($\sum F = 0$).

Linear Motion

• Path: The total length of the motion of an object. This is a scalar quantity, often denoted by $d$, $x$ (horizontal), or $y$ (vertical), measured in meters ($m$).

• Displacement ($\Delta d$, $\Delta x$, $\Delta y$): The straight-line distance from one point to another. This is a vector quantity and must include direction, measured in meters ($m$).

• Example Calculation: Given an initial point $P_1 = (300, 400)\,m$ and a final point $P_2 = (1200, 900)\,m$:

  • $\Delta x = 1200\,m - 300\,m = 900\,m$

  • $\Delta y = 900\,m - 400\,m = 500\,m$

  • Total displacement magnitude: $\sqrt{900^2 + 500^2} = 1,029\,m$

• Speed ($s$ or $v$): The rate at which an object changes its position (scalar).

• Velocity ($v$ or $\vec{v}$): The rate at which an object changes its position (vector). Units for both are $m/s$.

• Acceleration ($a$): The rate at which an object's velocity changes. It is a vector representing a speed up, slow down, or change in direction.

  • Equation: $a = \frac{\Delta v}{t} = \frac{v_{final} - v_{initial}}{t}$

  • Units: $m/s^2$

Kinematics and Dynamics Equations

• Kinematics: Focuses on "how" objects move under constant acceleration.

  • Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$

  • Average Acceleration: $a_{avg} = \frac{\Delta v}{\Delta t}$

  • Velocity-Time: $v_f = v_i + at$

  • Displacement-Time: $x_f = x_i + v_i t + \frac{1}{2} at^2$

  • Timeless Equation: $v_f^2 = v_i^2 + 2a\Delta x$

• Dynamics: Focuses on "why" things move.

  • Newton's Second Law: $\sum F = ma$

  • Force of Gravity (Weight): $F_g = mg$, where $g = 9.8\,m/s^2$ on Earth.

  • Friction Force: $F_f = \mu F_N$, where $\mu$ is the coefficient of friction and $F_N$ is the normal force.

Types of Forces

• Gravity: The most prevalent force, providing a constant downward pull toward the center of the Earth.

• Support Force (Normal Force, $F_N$): Occurs whenever an object is in contact with a surface (ground, floor, wall). It is perpendicular to the supporting surface.

• Friction: Resists motion between surfaces. Divided into two types:

  • Static Friction: The force required to get an object at rest to start moving. It is generally harder to start motion than to continue it.

  • Kinetic Friction: The force between two objects already in motion. The amount depends on the materials in contact (e.g., steel and ice have low friction; wood and steel have high friction) and the value of the support force ($more\,support\,force = more\,kinetic\,friction$).

• Air Resistance ($a.r.$): A force that occurs when an object travels through a fluid (liquid or gas), acting opposite to the direction of motion.

  • Air resistance is proportional to the speed of the object and depends on the form factor (shape).

  • Form Factor Example: A flat piece of paper takes longer to hit the ground than a crumpled piece because its larger form factor causes more drag.

  • Air resistance reduces acceleration. When gravity and air resistance become equal, the object reaches Terminal Speed (a state of dynamic equilibrium).

  • Terminal speed factors include size, shape, and density.

• Applied Force: A push or pull applied by an outside influence.

• Tension: The pulling force present in a string, cable, or cable system. It acts along the direction of the string.

• Elasticity/Spring: Quantifies the ability of objects to deform under force and return to their original configuration.

• Electromagnetism: Interactions between charged objects and magnetic materials. Like charges repel.

• Strong Force: Acts over very short distances to keep protons in a nucleus from repelling. It holds matter together and is involved in radioactive decay, fission, and fusion.

• Weak Force: Responsible for changing one type of quark into another (e.g., turning a proton into a neutron).

Newton's Second and Third Laws

• Newton's Second Law: $\sum F_{net} = ma$. The net force is the sum of all forces (gravity, support, friction, air resistance, etc.) acting on a mass.

  • Calculation Example: A box with $m = 0.35\,kg$ and $a = 0.9\,m/s^2$ has $\sum F = (0.35\,kg)(0.9\,m/s^2) = 0.315\,N$.

  • Friction Example: If a $25\,N$ force is applied to a heavy box that does not move, the net force is $0\,N$, meaning the static friction is $-25\,N$.

• Weight vs. Mass: Mass is constant everywhere (Earth, Moon, space). Weight varies by local gravity ($w = mg$).

  • Moon gravity is $\frac{1}{6}$ of Earth's.

  • Jupiter gravity is $2.5\,times$ heavier than Earth's.

  • Sun gravity is $30\,times$ heavier than Earth's.

  • In orbit, astronauts are weightless but still possess inertia/mass.

• Newton's Third Law: For every action, there is an equal and opposite reaction.

  • Forces always come in pairs. Any interaction between two objects involves action-reaction pairs that are equal in magnitude and opposite in direction, regardless of the objects' mass or speed.

  • Action-Reaction Confusion: Two forces acting on the same object cannot be action-reaction pairs. A pair requires two interacting objects.

• Questions & Discussion:

  • Q: How can anything move if all forces have an equal and opposite reaction?

  • A: Forces are equal and opposite, but the effect on motion (acceleration) is not equal because acceleration depends on mass ($F = ma$). Pushing off the floor involves the floor pushing back on you; you move because you have much less mass than the floor/Earth system.

Momentum and Impulse

• Momentum ($p$): Defined as "inertia in motion." It is a vector that quantifies motion.

  • Equation: $p = mv$

  • Units: $kg \cdot m/s$ or $Ns$

  • Mass and velocity are directly proportional to momentum. If direction differs between two objects, their momentums are not equal, regardless of mass and speed.

• Impulse (Change in Momentum, $\Delta p$): The change in momentum of an object or system.

  • Equation 1: $\Delta p = p_{final} - p_{initial}$

  • Equation 2: $\Delta p = F \Delta t$

  • Impulse is a vector and directly proportional to both applied force and the duration of time that force is applied.

  • Direction of Force: To speed up, apply force in the same direction as velocity; to slow down, apply force in the opposite direction.

• Conservation of Momentum: Momentum is conserved in the absence of external forces. This applies to all dimensions (x, y, and z independently).

Collisions and Energy

• Kinetic Energy ($KE$): Energy of motion ($KE = \frac{1}{2}mv^2$), measured in Joules ($J$).

• Inelastic Collision: Some kinetic energy is converted into other forms (heat, sound). Momentum is conserved.

• Perfectly Inelastic Collision: The maximum amount of kinetic energy is lost. The two objects collide and travel together as one unit.

• Elastic Collision: Kinetic energy is conserved. No heat or sound is generated, and molecules do not physically deform. Momentum is also conserved.

Work, Energy, and Power

• Energy: The capability of an object to do useful work. Concepts of work and energy allow for the analysis of simple machines, projectiles, planets, and roller coasters.

• Work ($W$): The transfer of energy by applying a force over a distance ($W = F_{\parallel}d$).

• Units of Work/Energy: Joule ($J$). $1\,J = 1\,Nm = 1\,kg \cdot m^2/s^2$.

  • Note: While $Nm$ is mathematically equivalent to a Joule, $Nm$ is specifically reserved for torque, not energy.

• Work-Energy Theorem: $W = \Delta KE = KE_{final} - KE_{initial}$.

  • Positive Work: Parallel force is in the same direction as displacement; the object speeds up (gains $KE$).

  • Negative Work: Parallel force is in the opposite direction of displacement; the object slows down (loses $KE$).

• Scalar Nature: Energy and work are scalars (no direction).

• Power ($P$): The rate at which work is done.

  • Equation: $P = W/t$

  • Units: Watt ($W$), where $1\,W = 1\,J/s$.

• Mechanical Energy ($ME$): The sum of potential and kinetic energy ($ME = GPE + KE$). It is conserved in the absence of dissipative forces like friction or air drag.

Forms of Potential Energy (Stored Energy)

• Gravitational Potential Energy (GPE): Comes from gravitational attraction between Earth and an object ($GPE = mgh$).

• Electric Potential Energy: Comes from the arrangement of charged particles.

• Chemical Energy: Comes from the arrangement of atoms or molecules.

• Nuclear Energy: Comes from the nucleus of an atom.

• Elastic Potential Energy: Comes from the configuration of a deformable object.

• Heat: Energy that flows due to temperature difference. Often considered wasted energy (e.g., friction converts $KE$ to heat).

Simple Machines

• Function: Change the direction, magnitude, or both of an applied force.

• Types: Levers, inclined planes, wheels and axles, wedges, screws, pulleys.

• Mechanical Advantage (MA): The multiplication of an input force by a machine.

  • Equation: $MA = \frac{F_{out}}{F_{in}}$

  • Units: None (it is a ratio).

Rotational Motion

• Translational Motion: Motion along a straight line.

• Rotational Speed: The speed at which an object rotates about an axis.

  • Measurement Units: Rotations/revolutions per minute (RPM), degrees per second ($^{\circ}/s$), or radians per second ($rad/s$). One rotation equals $2\pi\,radians$.

• Tangential Velocity ($v_t$): Relates rotational speed to the distance from the center ($r$).

  • Equation: $v_t = \frac{2\pi r}{t}$

  • Velocity is tangent to the path. Points further from the center have higher tangential speed; points closer to the center have lower speed.

• Rotational Acceleration: Measures the change in rotational speed.

• Rotational Inertia ($I$): Resistance to changes in rotational motion. It depends on the mass of the object and how that mass is distributed. Higher mass or mass distributed further from the axis makes it harder to change rotational motion.

• Torque (Symbol: $\tau$): A twist or turn causing change in rotational motion.

  • Equation: $\tau = F \cdot r$

  • Units: $Nm$

  • Torque is directly proportional to the applied force and the distance ($r$) from the center of rotation.

Gravity and Apparent Weight

• Apparent Weight (AW): The support force acting on a person. It is not necessarily constant ($AW = F_{support}$).

• Equation: $ma = F_{support} + mg$

• Elevator Scenarios:

  • Constant Speed: Acceleration is zero. $F_{support} = mg$. Net force is zero.

  • Speeding Up Upward: Net force points up. AW > mg.

    • Example: Person ($75\,kg$) accelerating up at $4\,m/s^2$.

    • $w = (75\,kg)(-9.8\,m/s^2) = -735\,N$

    • $\sum F = (75\,kg)(4\,m/s^2) = 300\,N_{up}$

    • $300\,N = F_{support} + (-735\,N) \rightarrow F_{support} = 1035\,N$.

  • Speeding Up Downward: Net force points down. AW < mg.

    • Example: Person ($75\,kg$) accelerating down at $-3\,m/s^2$.

    • $\sum F = (75\,kg)(-3\,m/s^2) = -225\,N$

    • $-225\,N = F_{support} + (-735\,N) \rightarrow F_{support} = 510\,N$.

• Free-Fall: If the elevator cable snaps and moves down at $9.8\,m/s^2$, apparent weight is zero.

Universal Gravitation

• Inverse-Square Law: Physical properties (Light, Sound, Gravity, Electromagnetism) dilute in 3D space by $\frac{1}{d^2}$. If distance doubles, the property decreases to one-quarter ($\frac{1}{4}$). If distance is halved, the property increases four times.

• Newton's Law of Universal Gravitation: Quantifies the force between two masses.

  • Equation: $F = G \frac{m_1 m_2}{d^2}$

  • Universal Gravitation Constant ($G$): $6.67 \times 10^{-11}\,N\cdot m^2/kg^2$.

  • Gravity is the weakest of the four fundamental forces and is always attractive because mass is always positive.

  • Force is only zero at infinite distance; therefore, all masses in the universe exert gravitational forces on each other.

• Example: Earth-Sun Force:

  • $m_E = 6 \times 10^{24}\,kg$, $m_S = 2 \times 10^{30}\,kg$, $d = 148 \times 10^9\,m$

  • $F = \frac{(6.67 \times 10^{-11})(6 \times 10^{24})(2 \times 10^{30})}{(148 \times 10^9)^2} = 3.65 \times 10^{20}\,N$ (This force keeps Earth in orbit).

• Gravitational Acceleration (Little g):

  • On Earth's surface: $g = -9.8\,m/s^2$.

  • Derived from $ma = G \frac{M_{earth} m_{object}}{r^2}$, giving $g = G \frac{M_{earth}}{r^2}$.

  • Earth Values: $M_{earth} = 5.97 \times 10^{24}\,kg$, $r_{earth} = 6.37 \times 10^6\,m$.

  • Example: Mt. Everest ($8,850\,m$ elevation):

    • $g = \frac{(6.67 \times 10^{-11})(5.97 \times 10^{24})}{(6.37 \times 10^6 + 8,850)^2} = 9.79\,m/s^2$.

• Gravitational Fields: A model using arrows to show where a mass would go due to gravity.

General Relativity

• Newton's Laws describe most motion accurately but fail for objects traveling close to the speed of light.

• Theory of General Relativity: Motion is influenced by curves in "spacetime." More massive objects cause greater curves.

Projectile and Satellite Motion

• Projectile Motion: Motion in free-fall (gravity only, ignoring friction/air drag). The path is a parabola.

• Ballistics: The study of objects briefly powered then moving only under gravity.

• Launch Angles and Horizontal Range:

  • 90 degrees: Maximum time in air; minimum horizontal motion.

  • 5 degrees: Minimum time in air; minimum horizontal motion.

  • 45 degrees: Perfect balance for maximum horizontal range.

• Solving Projectile Motion:

  • Use Kinematics ($d = \frac{1}{2}at^2 + v_i t$) for timing.

  • Use Conservation of Energy ($GPE + KE = constant$) for height.

  • Motion is symmetric: time to go up equals time to go down.

• Example Calculation:

  • $v_{x,i} = 10\,m/s$, $v_{y,i} = 30\,m/s$.

  • Total time: From $v_f = v_i + at \rightarrow -30 = 30 + (-9.8)t \rightarrow t = 6.12\,s$.

  • Horizontal distance: $d = v_x t = (10)(6.12) = 61.2\,m$.

• Satellites: Projectiles "falling around" the Earth.

  • Orbital Velocity Equation: $v = \sqrt{\frac{G M_{earth}}{r_{earth} + height}}$.

  • Height calculation for $400\,km$ orbit: $v = 7,672\,m/s$.

• Orbits: Generally elliptical. In the special case of a circular orbit, orbital height stays constant, and $KE + GPE = constant$.

Definitions of Science and Knowledge

• Science: Systematic pursuit and organization of knowledge through questioning, observation, and controlled experiments.

• Scientific Method: The process used to answer questions about the universe.

• Hypothesis: A proposed, testable explanation for an observation.

• Theory: An explanation supported by multiple, repeated experiments.

• Quantum Physics: Physical laws describing matter and energy on a very small scale.