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

Circular MotionCircular 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

DefinitionsDefinitions

• Orbital motion: object circles an axis that does not pass through the object itself

DefinitionsDefinitions

• Radius• Diameter• Chord• Tangent• Arc

Circle TerminologyCircle Terminology

Establishing PositionEstablishing Position• The simplest coordinate

system to use for circular motion puts the tails of position vectors at the center of the circular motion.

Polar CoordinatesPolar Coordinates

• magnitude of r = radius of circular path

• θ = angle of rotation• θ is measured in radians

(r, θ)(r, θ)

Radian MeasureRadian 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 MeasureRadian Measure• There are exactly 2π

radians in one complete circle.

• Unit analysis:• 180° = π radians

Establishing PositionEstablishing 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 VelocitySpeed and Velocity

ω =ΔθΔt

• ω is a scalar quantity.• It is commonly expressed

as number of rotations or revolutions per unit of time.• Ex. “rpm”


ω =ΔθΔt

• If angular speed is constant, then the rotating object experiences uniform circular motion.


ω =ΔθΔt

• In the SI, the units are radians per second.

• Written as:


rads

or s-1

• The velocity vector of a particle in circular motion is tangent to the circular path.

• This velocity is called tangential velocity.


• The magnitude of the tangential velocity is called the tangential speed, vt.


vt = |vt|

• Another formula for tangential speed is:


vt =l

Δt• arclength l = r × Δθ

• average tangential speed:


vt =rΔθΔt

AccelerationAcceleration

• Linear motion:

• Circular motion:

a =ΔvΔt

a =vt

2

r

AccelerationAcceleration• The instantaneous

acceleration vector always points toward the center of the circular path.

• This is called centripetal acceleration.

AccelerationAcceleration• The magnitude of

centripetal acceleration is:

ac =vt

2

rm/s²

• For all circular motion at constant radius and speed

AccelerationAcceleration• Another formula for

centripetal acceleration:

ac = -rω2

• Uniform angular velocity (ω) implies that the rate and direction of angular speed are constant.

Angular VelocityAngular Velocity

• Right-hand rule of circular motion:


• Nonuniform circular motion is common in the real world.

• Its properties are similar to uniform circular motion, but the mathematics are more challenging.


• change in angular velocity• notation: α• average angular

acceleration:

Angular AccelerationAngular Acceleration

α =ΔωΔt

ω2 – ω1 Δt

=

• units are rad/s², or s-2 • direction is parallel to the

rotational axis

Angular AccelerationAngular Acceleration

α =ΔωΔt

ω2 – ω1 Δt

==

• defined as the time-rate of change of the magnitude of tangential velocity

Tangential AccelerationTangential Acceleration

• average tangential acceleration:


at =Δvt

Δt= αr

• instantaneous tangential acceleration:


at = αr

Don’t be too concerned about the calculus involved here...

• Instantaneous tangential acceleration is tangent to the circular path at the object’s position.


• If tangential speed is increasing, then tangential acceleration is in the same direction as rotation.


• If tangential speed is decreasing, then tangential acceleration points in the opposite direction of rotation.


• note the substitutions here:

Equations of Circular Motion

Equations of Circular Motion

Dynamics of Circular Motion

Dynamics of Circular Motion

• in circular motion, the unbalanced force sum that produces centripetal acceleration

• abbreviated Fc

Centripetal ForceCentripetal Force

• to calculate the magnitude of Fc:


Fc =mvt²

r

• Centipetal force can be exerted through:• tension• gravity


• the product of a force and the force’s position vector

• abbreviated: τ • magnitude calculated by the

formula τ = rF sin θ

TorqueTorque

• r = magnitude of position vector from center to where force is applied

• F = magnitude of applied force

TorqueTorque

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

TorqueTorque

τ = rF sin θ

• Maximum torque is obtained when the force is perpendicular to the position vector.

• Angular acceleration is produced by unbalanced torques.

TorqueTorque

• Zero net torques is called rotational equilibrium.

• Στ = 0 N·m

TorqueTorque

• Law of Moments: l1F1 = l2F2 • Rearranged:

TorqueTorque

F1

F2

l2

l1

=

Universal GravitationUniversal

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 The Ideas

• Ptolemy developed a theory that involved epicycles in deferent orbits.

• For centuries, the geocentric view prevailed.

The Ideas The Ideas

• Copernicus concluded the geocentric theory was faulty.

• His heliocentric theory was simpler.

The Ideas The Ideas

• Tycho Brahe disagreed with both Ptolemy and Copernicus.

• He hired Johannes Kepler to interpret his observations.

The Ideas The Ideas

Kepler’s LawsKepler’s Laws• Kepler’s 1st Law states that

each planet’s orbit is an ellipse with the sun at one focus.

Kepler’s LawsKepler’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.

Kepler’s LawsKepler’s Laws

Kepler’s 2nd Law

Kepler’s LawsKepler’s Laws• Kepler’s 3rd Law relates the

size of each planet’s orbit to the time it takes to complete one orbit.

= KR³T²

Kepler’s LawsKepler’s Laws• R = length of semi-major

axis• T = time to complete one

orbit (period)

= KR³T²

Kepler’s LawsKepler’s Laws• R is measured in ua

(astronomical units), the mean distance from earth to the sun

= KR³T²

Kepler’s LawsKepler’s Laws• T is measured in years

= KR³T²

NewtonNewton• determined that gravity

controls the motions of heavenly bodies

• determined that the gravitational force between two objects depends on distance and mass

NewtonNewton• derived the Law of

Universal Gravitation:

Fg = GFg = G r²Mm

• G is called the universal gravitational constant

• Newton did not calculate G.

Law of Universal Gravitation


Fg = G r²Mm

• It predicts the gravitational force, but does not explain how it exists or why it works.



Fg = G r²Mm

• It is valid only for “point-like masses.”

• Gravity is always an attractive force.



Fg = G r²Mm

• Cavendish eventually determined the value of G through experimentation with a torsion balance.



Fg = G r²Mm

• G ~ 6.674 × 10-11 N·m²/kg²• Cavendish could then

calculate the mass and density of planet Earth.

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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