Energy and Universal Gravitation Lecture Review

Energy and Work

• Conceptual Foundation of Energy: Energy is directly related to the concept of work. It shares the exact same units of measurement.

• Definition of Work: Work is defined as the product of force and distance.     * Equation: $W = F imes d$     * Units: The unit for work is the Newton-meter ($N\cdot m$), which is equivalent to the Joule ($J$).

• Energy Units: Since energy is the capacity to do work, it is also measured in Joules ($J$).

Gravitational Potential Energy ($PE$)

• Definition: Often referred to as gravitational potential energy, it represents the energy stored in an object due to its position relative to a gravitational field.

• Calculation: Potential energy is the product of mass, gravity, and height.     * Equation: $PE = mgh$     * Mechanical Logic: The component $m \times g$ represents the force needed to lift an object against gravity, and $h$ represents the distance (height) the object is lifted.

• Standard Gravity ($g$): For all calculations, the acceleration due to gravity on Earth is taken as $9.8\,m/s^2$.

• Example Problem 1: Lifting a $10\,kg$ object to a height of $20\,meters$.     * Calculation: $10\,kg \times 9.8\,m/s^2 \times 20\,m = 1,960\,J$     * Result: The potential energy is $1,960\,joules$.

• Varieties of Potential Energy:     * Elastic Potential Energy: Stored in objects like springs (mentioned as appearing in previous problems but not on the final exam).     * Electrical Potential: In electricity, voltage is essentially potential energy per charge (work done per charge).

Kinetic Energy ($KE$)

• Definition: Kinetic energy is the energy possessed by an object in motion. Unlike potential energy, which has various forms, kinetic energy has only one standardized version.

• Equation: $KE = \frac{1}{2}mv^2$

• Unit Analysis: The combination of $\frac{1}{2} \times kg \times (m/s)^2$ results in $(kg\cdot m/s^2) \times m$, which is equivalent to Force ($N$) $\times$ distance ($m$), yielding Joules ($J$).

• Example Problem 2: A $15\,kg$ object moving at $25\,m/s$.     * Calculation: $0.5 \times 15\,kg \times (25\,m/s)^2 = 4,687.5\,J$     * Note: Gravity is irrelevant for kinetic energy equations involving straight-line horizontal movement.

Conservation of Energy

• Energy Transfer: Energy is transferred between potential and kinetic forms.     * Falling Objects: Potential energy at the top converts into kinetic energy at the bottom.     * Projectiles: Kinetic energy at the launch point converts into potential energy at the peak height.

• Equivalency Equation: At the point where no energy is lost to heat or friction, the maximum potential energy equals the maximum kinetic energy.     * Equation: $mgh = \frac{1}{2}mv^2$     * Simplified Formulas: Since mass occurs on both sides, it cancels out ($gh = \frac{1}{2}v^2$), allowing for the derivation of path-independent variables:         * Finding Height ($h$): $h = \frac{v^2}{2g}$         * Finding Velocity ($v$): $v = \sqrt{2gh}$

• Example Problem 3 (Potato Gun): A $1.2\,kg$ potato is shot straight up at $180\,m/s$.     * Calculation for Height: $h = \frac{(180\,m/s)^2}{2 \times 9.8\,m/s^2} = \frac{32,400}{19.6} = 1,653\,meters$     * Verification: An object at $1,653\,m$ falling back to Earth will strike the ground at a velocity of $180\,m/s$, determined by $v = \sqrt{2 \times 9.8 \times 1,653}$.

• Complex Scenarios: For an object already in motion that then changes elevation (e.g., a diving airplane), one must calculate the potential energy gained or lost and add/subtract it from the initial kinetic energy.

Historical Views of the Universe

• Geocentric Model (Earth-Centered):     * Concept: Proposed primarily by Ptolemy. It posited that the Earth was stationary and the center of the universe.     * Evidence: Observed stars rotating around the Earth daily. The Earth was perceived as immobile because movement could not be felt.     * Celestial Structure: Stars were believed to be fixed on a rotating crystal sphere.     * Spherical Earth: Educated individuals (dating back to before the time of Christ) were well-aware the Earth was spherical. The notion that people like Columbus feared sailing off a flat edge is historically inaccurate.

• The Problem of "Wanderers" (Planets):     * Planets did not follow the uniform circular paths of stars.     * Retrograde Motion: Planets appear to move across the sky, turn back, and then move forward again.     * Epicycles: To fix the geocentric model, astronomers proposed "cycles upon cycles." This required constant manual adjustment to maintain predictive accuracy.

• Heliocentric Model (Sun-Centered):     * Copernicus: Proposed that the Sun is the center and planets move in perfect circles. He correctly identified Earth as the third planet from the Sun.     * Flaws in Copernicus's Model: It was no more accurate at predicting planet locations than the geocentric model because it relied on perfect circles. It also lacked a physical explanation for why human beings weren't blown off the Earth by the rotational speed (calculated at approximately $1,000\,miles/hour$ based on a circumference of $24,000\,miles$ rotating every $24\,hours$).

Kepler’s Laws of Planetary Motion

• Johannes Kepler: Refined the heliocentric model by replacing perfect circles with elliptical paths.

• First Law (Law of Orbits): The paths of the planets are ellipses with the Sun at one focus.

• Second Law (Law of Areas): A planet sweeps out equal areas in equal periods of time.     * Implication: Planets speed up as they get closer to the Sun and slow down as they move further away.     * Example (Halley's Comet): Has an $86\text{-year}$ orbit. It moves incredibly fast near the Sun (developing a tail) and very slowly at the outer edges of its orbit.

• Third Law: The ratio of the cube of the radius ($r$) to the square of the period ($T$) is a constant for any object orbiting a specific central body.     * Equation: $\frac{r^3}{T^2} = \text{constant}$     * Newton later used this constant to derive laws relating to force.

Newton’s Universal Law of Gravitation

• Derivation: Newton linked Kepler's third law with centripetal force equations ($F_c = \frac{4\pi^2r}{T^2}$) to realize the "constant" was actually the gravitational pull of the Sun.

• The Law: Every object in the universe attracts every other object with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.     * Equation: $F = G \frac{m_1 m_2}{r^2}$

• Universal Gravitational Constant ($G$): $G = 6.67 \times 10^{-11}\,N\cdot m^2/kg^2$. Although tiny, it creates significant force when applied to the massive scales of the solar system.

• Inverse Square Law: Moving an object twice as far away ($2r$) reduces the force to one-quarter ($\frac{1}{4}$). Moving it three times as far ($3r$) reduces the force to one-ninth ($\frac{1}{9}$).

• Newton's Verification: Newton tested this by comparing the gravity at Earth's surface ($9.8\,m/s^2$) with the Moon's motion. Knowing the Moon is at a distance of $60\,Earth\,radii$, its centripetal acceleration should be (and is) approximately $\frac{9.8}{60^2}$.