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Circular Motion. Definitions. Circular motion: when an object moves in a two-dimensional circular path Spin: object rotates about an axis that pass through the object itself. Definitions. Orbital motion: object circles an axis that does not pass through the object itself.
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Definitions • Circular motion: when an object moves in a two-dimensional circular path • Spin: object rotates about an axis that pass through the object itself
Definitions • Orbital motion: object circles an axis that does not pass through the object itself
Circle Terminology • Radius • Diameter • Chord • Tangent • Arc
Establishing Position • The simplest coordinate system to use for circular motion puts the tails of position vectors at the center of the circular motion.
Polar Coordinates (r, θ) • magnitude of r = radius of circular path • θ = angle of rotation • θ is measured in radians
Radian Measure • Definition of a radian: One radian is equal to the central angle of a circle that subtends an arc of the circle’s circumference whose length is equal to the length of the radius of the circle.
Radian Measure • There are exactly 2π radians in one complete circle. • Unit analysis: • 180° = π radians
Establishing Position • In circular motion, change of position is measured in angular units. • θ can have a positive or negative value.
Δθ ω = Δt Speed and Velocity • ω represents the time-rate of change of angular position; this is also called the angular speed. • By definition:
Δθ ω = Δt Speed and Velocity • ω is a scalar quantity. • It is commonly expressed as number of rotations or revolutions per unit of time. • Ex. “rpm”
Δθ ω = Δt Speed and Velocity • If angular speed is constant, then the rotating object experiences uniform circular motion.
rad or s-1 s Speed and Velocity • In the SI, the units are radians per second. • Written as:
Speed and Velocity • The velocity vector of a particle in circular motion is tangent to the circular path. • This velocity is called tangential velocity.
Speed and Velocity • The magnitude of the tangential velocity is called the tangential speed, vt. vt = |vt|
l vt = Δt Speed and Velocity • Another formula for tangential speed is: • arclengthl = r × Δθ
rΔθ vt = Δt Speed and Velocity • average tangential speed:
Δv a = Δt vt2 a = r Acceleration • Linear motion: • Circular motion:
Acceleration • The instantaneous acceleration vector always points toward the center of the circular path. • This is called centripetal acceleration.
vt2 ac = m/s² r Acceleration • The magnitude of centripetal acceleration is: • For all circular motion at constant radius and speed
Acceleration • Another formula for centripetal acceleration: ac = -rω2
Angular Velocity • Uniform angular velocity (ω) implies that the rate and direction of angular speed are constant.
Angular Velocity • Right-hand rule of circular motion:
Angular Velocity • Nonuniform circular motion is common in the real world. • Its properties are similar to uniform circular motion, but the mathematics are more challenging.
ω2 – ω1 Δω α = = Δt Δt Angular Acceleration • change in angular velocity • notation: α • average angular acceleration:
ω2 – ω1 Δω α = = Δt Δt Angular Acceleration • units are rad/s², or s-2 • direction is parallel to the rotational axis
Tangential Acceleration • defined as the time-rate of change of the magnitude of tangential velocity
Δvt at = =αr Δt Tangential Acceleration • average tangential acceleration:
Tangential Acceleration • instantaneous tangential acceleration: at =αr Don’t be too concerned about the calculus involved here...
Tangential Acceleration • Instantaneous tangential acceleration is tangent to the circular path at the object’s position.
Tangential Acceleration • If tangential speed is increasing, then tangential acceleration is in the same direction as rotation.
Tangential Acceleration • If tangential speed is decreasing, then tangential acceleration points in the opposite direction of rotation.
Equations of Circular Motion • note the substitutions here:
Centripetal Force • in circular motion, the unbalanced force sum that produces centripetal acceleration • abbreviated Fc
mvt² Fc = r Centripetal Force • to calculate the magnitude of Fc:
Centripetal Force • Centipetal force can be exerted through: • tension • gravity
Torque • the product of a force and the force’s position vector • abbreviated: τ • magnitude calculated by the formula τ = rF sin θ
Torque τ = rF sin θ • r = magnitude of position vector from center to where force is applied • F = magnitude of applied force
Torque τ = rF sin θ • θ = smallest angle between vectors r and F when they are positioned tail-to-tail • r sin θ is called the moment arm (l) of a torque
Torque • Maximum torque is obtained when the force is perpendicular to the position vector. • Angular acceleration is produced by unbalanced torques.
Torque • Zero net torques is called rotational equilibrium. • Στ= 0 N·m
F1 l2 = F2 l1 Torque • Law of Moments: l1F1 = l2F2 • Rearranged:
The Ideas • Geocentric: The earth is the center of the universe • Heliocentric: The sun is the center of the universe • Some observations did not conform to the geocentric view.
The Ideas • Ptolemy developed a theory that involved epicycles in deferent orbits. • For centuries, the geocentric view prevailed.
The Ideas • Copernicus concluded the geocentric theory was faulty. • His heliocentric theory was simpler.
The Ideas • Tycho Brahe disagreed with both Ptolemy and Copernicus. • He hired Johannes Kepler to interpret his observations.
Kepler’s Laws • Kepler’s 1st Law states that each planet’s orbit is an ellipse with the sun at one focus.
Kepler’s Laws • Kepler’s 2nd Law states that the position vector of a planet travels through equal areas in equal times if the vector is drawn from the sun.
