Circular Motion

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 Definitions

Radius Diameter Chord Tangent Arc Circle Terminology

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 magnitude of r = radius of circular path θ = angle of rotation θ is measured in radians (r, θ)

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.

ω represents the time-rate of change of angular position; this is also called the angular speed. By definition: Speed and Velocity ω = Δθ ΔtΔt

ω is a scalar quantity. It is commonly expressed as number of rotations or revolutions per unit of time. Ex. “rpm” Speed and Velocity ω = Δθ ΔtΔt

If angular speed is constant, then the rotating object experiences uniform circular motion. Speed and Velocity ω = Δθ ΔtΔt

In the SI, the units are radians per second. Written as: Speed and Velocity rad s ors -1

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, v t. Speed and Velocity v t = | v t |

Another formula for tangential speed is: Speed and Velocity v t = l ΔtΔt arclength l = r × Δθ

average tangential speed: Speed and Velocity v t = rΔθ ΔtΔt

Acceleration Linear motion: Circular motion: a = ΔvΔv ΔtΔt vt2vt2 r

Acceleration The instantaneous acceleration vector always points toward the center of the circular path. This is called centripetal acceleration.

Acceleration The magnitude of centripetal acceleration is: a c = vt2vt2 r m/s² For all circular motion at constant radius and speed

Acceleration Another formula for centripetal acceleration: a c = - r ω 2

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. Angular Velocity

change in angular velocity notation: α average angular acceleration: Angular Acceleration α = Δω ΔtΔt ω 2 – ω 1 ΔtΔt =

units are rad/s², or s -2 direction is parallel to the rotational axis Angular Acceleration α = Δω ΔtΔt ω 2 – ω 1 ΔtΔt = =

defined as the time-rate of change of the magnitude of tangential velocity Tangential Acceleration

average tangential acceleration: Tangential Acceleration a t = ΔvtΔvt ΔtΔt = αr= αr

instantaneous tangential acceleration: Tangential Acceleration a t = α r Don’t be too concerned about the calculus involved here...

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. Tangential Acceleration

note the substitutions here: Equations of Circular Motion

Dynamics of Circular Motion

in circular motion, the unbalanced force sum that produces centripetal acceleration abbreviated F c Centripetal Force

to calculate the magnitude of F c : Centripetal Force F c = mv t ² r

Centipetal force can be exerted through: tension gravity Centripetal Force

the product of a force and the force’s position vector abbreviated: τ magnitude calculated by the formula τ = rF sin θ Torque

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 τ = rF sin θ

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. Σ τ = 0 N·m Torque

Law of Moments: l 1 F 1 = l 2 F 2 Rearranged: Torque F1F1 F2F2 l2l2 l1l1 =

Universal Gravitation

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. The Ideas

Kepler’s Laws Kepler’s 1 st Law states that each planet’s orbit is an ellipse with the sun at one focus.

Kepler’s Laws Kepler’s 2 nd Law states that the position vector of a planet travels through equal areas in equal times if the vector is drawn from the sun.

Kepler’s Laws Kepler’s 2 nd Law

Kepler’s Laws Kepler’s 3 rd Law relates the size of each planet’s orbit to the time it takes to complete one orbit. = K R³ T²

Kepler’s Laws R = length of semi-major axis T = time to complete one orbit (period) = K R³ T²

Kepler’s Laws R is measured in ua (astronomical units), the mean distance from earth to the sun = K R³ T²

Kepler’s Laws T is measured in years = K R³ T²

Newton determined that gravity controls the motions of heavenly bodies determined that the gravitational force between two objects depends on distance and mass

Newton derived the Law of Universal Gravitation: F g = G r² Mm G is called the universal gravitational constant Newton did not calculate G.

Law of Universal Gravitation F g = G r² Mm It predicts the gravitational force, but does not explain how it exists or why it works.

Law of Universal Gravitation F g = G r² Mm It is valid only for “point- like masses.” Gravity is always an attractive force.

Law of Universal Gravitation F g = G r² Mm Cavendish eventually determined the value of G through experimentation with a torsion balance.

Law of Universal Gravitation F g = G r² Mm G ~ × N·m²/kg² Cavendish could then calculate the mass and density of planet Earth.