Gravity, Radiation Biology, and Rotational Kinematics: PHYS 111 Study Notes

Gravity and Newton's Law of Gravitation

Inverse Square Law for Gravity

Gravity is a fundamental force that acts between any two masses in the universe. Its strength decreases with the square of the distance between the centers of the masses.

• Newton's Law of Universal Gravitation: The gravitational force between two masses is given by: $F = G \frac{m_1 m_2}{R^2}$ where G is the gravitational constant ($6.674 \times 10^{-11} \text{N m}^2/\text{kg}^2$), m1 and m2 are the masses, and R is the distance between their centers.

• Vector Nature: Gravitational forces are vectors, meaning they have both magnitude and direction.

• Equal and Opposite Forces: The force exerted by mass 1 on mass 2 is equal in magnitude and opposite in direction to the force exerted by mass 2 on mass 1.

• Distance Measurement: The distance R is always measured from the center of mass of each object, not from their surfaces.

Example Calculation

Given two masses, 100 kg and 75 kg, separated by 2 meters, the gravitational force between them is:

• $F = G \frac{100 \times 75}{2^2}$

Gravitational Potential Energy

Potential energy in a gravitational field is also governed by an inverse-square law.

• Formula: $U = -\frac{G m_1 m_2}{r}$

• Total Mechanical Energy: The total energy of an orbiting body is the sum of its kinetic and potential energy.

• Bound and Unbound States: If the total energy is negative, the object is gravitationally bound; if zero or positive, it is unbound.

Why Study Gravity?

• Gravity serves as an analog for electricity and magnetism due to its inverse-square law nature.

• It is a testing ground for conservation of energy and angular momentum.

• Gravitational models mimic molecular interactions in physics and chemistry.

• Gravitational potential is analogous to electric potential.

• Quantum mechanics and gravity impact planetary studies and space travel.

Kepler's Laws of Planetary Motion

Overview of Kepler's Laws

Kepler's Laws describe the motion of planets and other celestial bodies in orbit.

• First Law (Law of Ellipses): Planetary orbits are ellipses, with the Sun at one focus.

• Second Law (Law of Equal Areas): A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. This is a consequence of conservation of angular momentum.

• Third Law (Law of Periods): The square of the orbital period (T) is proportional to the cube of the semi-major axis (R): $T^2 \propto R^3$ or $\frac{T^2}{R^3} = \text{constant}$

Application Example

• If the period of Mercury is 88 days and the period of Earth is 365.25 days, the ratio of their orbital radii can be found using: $\left(\frac{T_2}{T_1}\right)^2 = \left(\frac{R_2}{R_1}\right)^3$

Derivation of Kepler's Third Law

• For a circular orbit: $a = -\frac{v^2}{r}$ $v = \frac{2\pi r}{T}$ $a = -\frac{(2\pi r / T)^2}{r}$ $a = -\frac{4\pi^2 r}{T^2}$ Setting gravitational force equal to centripetal force: $\frac{GM}{r^2} = \frac{4\pi^2 r}{T^2}$ Rearranging: $T^2 = \frac{4\pi^2}{GM} r^3$ Thus, $T^2 / r^3 = \text{constant}$

Escape Velocity

Definition and Formula

Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a massive body.

• Formula: $E = \frac{1}{2}mv^2 - \frac{GmM}{r}$

• Setting total energy to zero and solving for v gives the escape velocity.

• If the total energy is negative, the object remains bound; if zero or positive, it escapes.

Gravity as a Vector and Gauss' Law Analogy

Superposition Principle

When multiple bodies exert gravitational forces, the net force is the vector sum of all individual forces.

• Gravitational force is always attractive.

• Forces between pairs are equal and opposite.

• At the center of a planet, the net gravitational force is zero due to symmetry.

• Only the mass inside a given radius contributes to the gravitational force at that radius (Gauss' Law analogy).

Radiation Biology

Primary Effects of Radiation

Radiation can cause significant biological effects, especially at the molecular and cellular level.

• DNA Damage: Radiation can break chemical bonds in DNA, leading to mutations or cell death.

• Water Ionization: Radiation splits water into H+ and OH-, which can further damage biomolecules.

• Cellular Repair: The body has mechanisms to repair radiation-induced damage, but excessive exposure can overwhelm these systems.

• Apoptosis: Cells may be programmed to die if damage is irreparable.

Energy Deposition and Dose

• Linear No-Threshold Model: Assumes any amount of radiation has some risk of causing cancer.

• Dose: The amount of energy deposited per unit volume of tissue.

• High vs. Low Dose: Large doses over short times are more harmful; lower doses over longer times allow for adaptation and repair.

Medical Imaging and Therapy

• Imaging: Medical imaging uses low doses of radiation, which have not been shown to cause cancer in studies so far.

• Therapy: Radiation therapy aims to deliver high doses to cancer cells while sparing healthy tissue.

• Technological Advances: New technologies improve the signal-to-noise ratio, reducing unnecessary exposure.

Common Medical Radiation Exposures

Procedure	Typical Dose (mSv)
Chest X-ray	Low
Mammogram	Low
Dental Bite-wing	Very Low
CT Chest	Moderate
Coronary Angiography	Higher
PET/CT	Higher
Lung Perfusion/Ventilation	Moderate
Gastrointestinal (nuclear medicine)	Moderate
Additional info: Actual dose values depend on procedure and patient size.

Rotational Kinematics

Connection Between Linear and Rotational Variables

Rotational kinematics studies the motion of objects as they rotate about an axis, drawing analogies to linear motion.

• Arc Length and Angle: The arc length s is related to the angle θ (in radians) by $s = r\theta$.

• Radians vs. Degrees: Radians are the standard unit for angular measurements in physics.

• Angular Velocity and Acceleration:

  • Angular velocity: $\omega = \frac{d\theta}{dt}$

  • Angular acceleration: $\alpha = \frac{d\omega}{dt}$

• Analogies to Linear Kinematics:

  • Linear displacement $x$ → angular displacement $\theta$

  • Linear velocity $v$ → angular velocity $\omega$

  • Linear acceleration $a$ → angular acceleration $\alpha$

• Connections Between Accelerations:

  • Tangential acceleration: $a_t = r\alpha$

  • Centripetal acceleration: $a_c = \omega^2 r$

• Rolling Motion: Rolling combines rotational and translational motion.

Right-Hand Rule

Angular quantities are vectors. The direction of angular velocity or angular acceleration is given by the right-hand rule: curl the fingers in the direction of rotation, and the thumb points along the axis of rotation.

Rotational Dynamics

Rotational motion is governed by analogs of Newton's Laws, with torque playing the role of force.

• Torque: $\tau = rF\sin\theta$

• Moment of Inertia: $I = \sum m_i r_i^2$

• Rotational Newton's Law: $\tau = I\alpha$

Additional info: These principles are foundational for understanding planetary motion, mechanical systems, and biological processes involving rotation.