Circular Motion

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# Circular Motion - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Circular Motion' - jenn

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Presentation Transcript

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

• 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

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.

• There are exactly 2π radians in one complete circle.
• Unit analysis:

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.

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.

Kepler’s Laws

Kepler’s 2nd Law

= K

Kepler’s Laws

• Kepler’s 3rd Law relates the size of each planet’s orbit to the time it takes to complete one orbit.

= K

Kepler’s Laws

• R = length of semi-major axis
• T = time to complete one orbit (period)

= K

Kepler’s Laws

• R is measured in ua (astronomical units), the mean distance from earth to the sun

= K

Kepler’s Laws

• T is measured in years

Newton

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

Mm

Fg = G

Newton

• derived the Law of Universal Gravitation:
• G is called the universal gravitational constant
• Newton did not calculate G.

Mm

Fg = G

Law of Universal Gravitation

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

Mm

Fg = G

Law of Universal Gravitation

• It is valid only for “point-like masses.”
• Gravity is always an attractive force.

Mm

Fg = G

Law of Universal Gravitation

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

Mm

Fg = G

Law of Universal Gravitation

• G ~ 6.674 × 10-11 N·m²/kg²
• Cavendish could then calculate the mass and density of planet Earth.