Study Notes on Circular Motion, Orbits & Gravity

MOTION IN A CIRCLE

Learning Outcomes

• At the end of this chapter you should be able to:

  • Apply kinematics and dynamics knowledge, skills, and techniques to circular motion.

Mechanics Overview

• Chapter 6 covers Circular Motion, Orbits & Gravity.

Circular Motion Fundamentals

Particle Model

• In the particle model, the center of the circle lies outside the particle, leading to the discussion of orbital motion.

• This same principle applies later to the rotation or spin of extended objects about their own axes.

Uniform Circular Motion

• Definition: A particle traveling at constant speed around a circle enters into uniform circular motion.

  • The magnitude of velocity is constant.

  • The velocity vector is always tangent to the circle, indicating that its direction changes continuously.

Period of Motion

Definition of Period (T)

• The time taken for the particle to complete one revolution (rev) is called the period (T) of the motion.

Angular Position

Characteristics of Angular Position (θ)

• Angular Position is measured as:

  • Positive when measured counterclockwise (ccw) from the positive x-axis.

  • Conveniently measured in radians (SI unit).

• It is the single time-dependent quantity of circular motion.

• Polar coordinates are preferred over Cartesian coordinates (xy-coordinates) when describing the position of orbiting particles.

Radians and Angle Measurement

Notes on Radians

• The radian is a dimensionless unit, applicable to any unit of angle.

Angular Velocity

Definition and Calculation

• The change in angular position is called angular displacement.

• The average angular velocity can be defined as:

  • $ext{Average Angular Velocity} = rac{ ext{Angular Displacement}}{ ext{Time}}$

• For instantaneous angular velocity, this becomes:

  • $ext{Angular Velocity} = rac{d heta}{dt}$

• Units of angular velocity include:

  • [rad/s] (SI unit)

  • [°/s, rev/s, rev/min (rpm)]

Angular Velocity Conditions

• A particle moves with uniform circular motion if and only if its angular velocity is constant.

• Angular velocity is positive for counterclockwise motion and negative for clockwise motion.

• Graphical relationships developed for position and velocity in linear motion apply equally well to angular position and angular velocity.

Position and Velocity Graphs

Example of a Particle in Motion

• A particle describes its position through POSITION GRAPHS, which relate to VELOCITY GRAPHS.

• Angular velocity can be interpreted as the slope of these graphs.

Circular Motion Dynamics

Characteristics of Circular Motion

• Uniform circular motion consists of constant speed but with continuously changing direction, which indicates accelerated motion.

• The period (T) is the time to complete one revolution around a circular trajectory.

• The frequency (f) represents the number of revolutions per second.

Centripetal Acceleration

Definition

• Centripetal acceleration is directed towards the center of the circular path.

• Identified through its relationship with motion in a circle.

Notes on Circular Motion

• When analyzing particles in circular motion, their instantaneous velocity and acceleration vectors are at right angles to each other.

• Whenever there is motion in a circle, centripetal acceleration is always present, even when speed is not constant.

Dynamics of Uniform Circular Motion

Newton's Second Law in Circular Motion

• The net force acting on any body moving in circular motion is the result of identifiable forces applied towards the center of the circle. It’s crucial to remember that this is NOT a new, physical force, but a result of existing forces.

Apparent Forces in Circular Motion

Explanation of Apparent Forces

• As an object moves in a circle, an apparent force seems to push it outward; however, this force does not exist in reality—it is merely a sensation.

Real Forces at Play

• The centrifugal force is defined as an apparent force experienced when moving in a curved path.

Banked Curves in Motion

Dynamics of Banked Curves

• On a banked curve, the angle at which the road is inclined affects the maximum speed a vehicle can navigate the corner without relying on friction.

• This consideration is essential for understanding vehicle dynamics on curved roads.

Motion in Vertical Circles

Dynamics of Vertical Circular Motion

• Motion in a vertical circle is NOT uniform due to the constant gravitational force acting downwards.

• The relationship between tension forces and weight change between points at the top versus the bottom of the vertical circle:

  • At the top, tension force and weight are aligned (both acting downwards).

  • At the bottom, tension and weight are anti-parallel (tension acts upwards while weight acts downwards).

Apparent Weight

• At the bottom of a vertical circle, the sensation of apparent weight is a result of the contact forces providing support, rather than being a reflection of the gravitational force.

• The equation governing apparent weight can be expressed as:

  • $F_{ ext{n}} = m rac{v^2}{r} + mg$ (where ${F}_{n}$ is the normal force, ${v}$ is velocity, ${m}$ is mass, ${g}$ is acceleration due to gravity, and ${r}$ is the radius).

Critical Speed

• The critical speed occurs when normal force necessitates maintaining a minimum speed to avoid free fall.

• The necessary equation defining critical speed is:

  • $v_c = rac{g r}{ ext{c}}$, where c defines boundary conditions for forces in play.

Circular Orbits

Nature of Circular Orbits

• The force maintaining a satellite's circular orbit around a planetary body, such as the Earth, is indeed the gravitational forces acting between them.

Centripetal Acceleration in Orbits

• A near-Earth satellite will sustain its circular orbit if its centripetal acceleration equals the gravitational acceleration due to Earth's mass:

  • $a_c = g$ where $g$ is the local gravitational acceleration.

Newton’s Law of Universal Gravitation

Definition

• Any two particles in the universe exert a force on each other that is:

  • Proportional to the product of their masses.

  • Inversely proportional to the square of the distance between their centers:

  • $F = G rac{m_1 m_2}{r^2}$

  • Where (G) is the universal gravitational constant.

Implications for Extended Masses

• This law also holds for extended spherical masses, such as planets, assuming that the distance between their centers is considerably greater than their sizes.

Weight and Free-Fall Acceleration

Weight Definition

• An object's weight is the force due to gravity at the surface of a planet, calculated based on its mass.

Gravitational Acceleration Formula

• The value of gravitational acceleration at the surface of any planet depends on its size and mass:

  • $g = G rac{M}{R^2}$ where M is the mass of the planet and R is its radius.

Earth Specifics

• For Earth, substituting gives values:

  • $g = 9.83 ext{ m/s}^2$ and adjustments observed yielding $g_{ ext{app}} = 9.80 ext{ m/s}^2$ based on other influences like rotation.

Variation of g with Height Above Ground

Gravitational Acceleration Changes by Altitude

• Heights and gravitational acceleration (g) values for different elevations:

  • Sea level (0 m): 9.83 m/s²

  • Kilimanjaro (5,900 m): 9.81 m/s²

  • Jet airliner (10,000 m): 9.80 m/s²

  • ISS (350,000 m): 8.85 m/s²

  • Geosynchronous satellite (35,900,000 m): 0.22 m/s²

Weightlessness Explained

Concept of Weightlessness

• Weightlessness is not the absence of gravitational force (g) but a condition arising from the absence of contact forces, indicating that the object is in free-fall.