circular motion. circular motion: when an object moves in a two- dimensional circular path spin:...
TRANSCRIPT
• 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
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”
Speed and VelocitySpeed and Velocity
ω =ΔθΔt
• If angular speed is constant, then the rotating object experiences uniform circular motion.
Speed and VelocitySpeed and Velocity
ω =ΔθΔt
• In the SI, the units are radians per second.
• Written as:
Speed and VelocitySpeed and Velocity
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.
Speed and VelocitySpeed and Velocity
• The magnitude of the tangential velocity is called the tangential speed, vt.
Speed and VelocitySpeed and Velocity
vt = |vt|
• Another formula for tangential speed is:
Speed and VelocitySpeed and Velocity
vt =l
Δt• arclength l = r × Δθ
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
• Uniform angular velocity (ω) implies that the rate and direction of angular speed are constant.
Angular VelocityAngular 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 VelocityAngular Velocity
• 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
• instantaneous tangential acceleration:
Tangential AccelerationTangential 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.
Tangential AccelerationTangential Acceleration
• If tangential speed is increasing, then tangential acceleration is in the same direction as rotation.
Tangential AccelerationTangential Acceleration
• If tangential speed is decreasing, then tangential acceleration points in the opposite direction of rotation.
Tangential AccelerationTangential Acceleration
• in circular motion, the unbalanced force sum that produces centripetal acceleration
• abbreviated Fc
Centripetal ForceCentripetal Force
• 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
• 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 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
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
Law of Universal Gravitation
Fg = G r²Mm
• It predicts the gravitational force, but does not explain how it exists or why it works.
Law of Universal Gravitation
Law of Universal Gravitation
Fg = G r²Mm
• It is valid only for “point-like masses.”
• Gravity is always an attractive force.
Law of Universal Gravitation
Law of Universal Gravitation
Fg = G r²Mm
• Cavendish eventually determined the value of G through experimentation with a torsion balance.