Circular MotionCircular Motion

Chapter 7 (already)Chapter 7 (already)

Polar CoordinatesPolar Coordinates

We commonly use We commonly use Cartesian or Cartesian or rectangular coordinate rectangular coordinate system where (x, y) system where (x, y) identifies a point in two identifies a point in two dimensions.dimensions.We could equivalently We could equivalently use polar coordinates use polar coordinates to identify a point (r, to identify a point (r, θθ))

Converting Between Coordinate Converting Between Coordinate SystemsSystems

Given r and Given r and θθ,,

X = r·cosX = r·cosθθ

Y = r·sinY = r·sinθθ

This converts from a This converts from a polar coordinate to a polar coordinate to a rectangular rectangular coordinatecoordinate

Given x and y,Given x and y,

r = (xr = (x22 + y + y22) ) ½½

θθ = tan = tan -1-1 (y/x) (y/x)

This converts from a This converts from a rectangular rectangular coordinate to a polar coordinate to a polar coordinatecoordinate

Circular MotionCircular Motion

Imagine a point Imagine a point moving along the moving along the circle at constant circle at constant speed. speed.

Which variables Which variables change and which change and which stay the same?stay the same?

Circular MotionCircular MotionThe angle, The angle, θθ, changes , changes but the radius does not but the radius does not change.change.

The coordinates x- and y- The coordinates x- and y- both change.both change.

The speed does not The speed does not change but the velocity change but the velocity does change (why?) does change (why?)

Angle MeasureAngle MeasureAngles may be Angles may be measured in degrees measured in degrees or in radians.or in radians.

360360° = 2° = 2ππ

The length of an arc The length of an arc subtending a circle of subtending a circle of radius r is given by radius r is given by

S = rS = r··θθ

Examples Examples

A spectator is standing at the center of a A spectator is standing at the center of a circular running track and observe a runner start circular running track and observe a runner start practice 256 m due east of her position. The practice 256 m due east of her position. The runner runs on the track in a counter clockwise runner runs on the track in a counter clockwise direction to the finish line, located due north of direction to the finish line, located due north of the observer. What was the distance of the run?the observer. What was the distance of the run?A sailor sights a distant ship and finds that it A sailor sights a distant ship and finds that it subtends an angle of 1.15subtends an angle of 1.15°. He knows from the °. He knows from the shipping charts that the tanker is 150 m in shipping charts that the tanker is 150 m in length. How far away is the tanker?length. How far away is the tanker?

Angular Speed / VelocityAngular Speed / VelocityAngular speed, Angular speed, ω, is defined as the rate of change ω, is defined as the rate of change of θ.of θ.

ω = Δθ/ Δt [radians/sec]ω = Δθ/ Δt [radians/sec]

andand θ = ωtθ = ωt

Then Then s = rθs = rθ becomes s = r (ωt) becomes s = r (ωt)

And And v = r ωv = r ω

Speeds in Circular MotionSpeeds in Circular Motion

Angular velocity is denoted with Angular velocity is denoted with ω, omega, ω, omega, and the direction is at a right angle to the and the direction is at a right angle to the plane of the circular motion (use right hand plane of the circular motion (use right hand rule)rule)

Tangential velocity is denoted with v, Tangential velocity is denoted with v, velocity.velocity.

Radians~!Radians~!

Note that Note that ω is measured in ω is measured in radiansradians/sec. /sec.

The formulas involving ω are true only if θ The formulas involving ω are true only if θ (angular displacement) is measured in (angular displacement) is measured in radians since only then is s = rθ a true radians since only then is s = rθ a true statement.statement.

Period and FrequencyPeriod and Frequency The Period, T, is the time necessary for one The Period, T, is the time necessary for one complete cycle. Period, T, is measured in complete cycle. Period, T, is measured in [seconds].[seconds].

The frequency, The frequency, ff, is the number of cycles that , is the number of cycles that happen in one second. Frequency is measured happen in one second. Frequency is measured in [cycles/second] or [Hertz]in [cycles/second] or [Hertz]

T = 1/T = 1/ff

Angular speed, Angular speed, ω = 2π/T = 2πω = 2π/T = 2πff

ExamplesExamples

A merry go round at its constant operational A merry go round at its constant operational speed makes one complete rotation in 45 speed makes one complete rotation in 45 seconds. Two children are on horses, on at 3.0 seconds. Two children are on horses, on at 3.0 m from the center and one 6.0 m from the m from the center and one 6.0 m from the center. Find the angular speed and the center. Find the angular speed and the tangential speed for each child.tangential speed for each child.A CD rotates in the player at a constant speed of A CD rotates in the player at a constant speed of 200 rpm. What are the CD’s a) frequency and b) 200 rpm. What are the CD’s a) frequency and b) Period of revolution?Period of revolution?

Centripetal AccelerationCentripetal Acceleration

An object in Uniform Circular Motion An object in Uniform Circular Motion (constant speed) experiences a (constant speed) experiences a Centripetal Force ( and therefore a Centripetal Force ( and therefore a centripetal acceleration) that is pointed centripetal acceleration) that is pointed towards the center.towards the center.

aacc = v = v22/r = (r/r = (rωω))22/r = r/r = rωω22

FFcc = m( = m(vv22/r) = mr/r) = mrωω22

ExamplesExamplesA lab centrifuge operates at a rotational A lab centrifuge operates at a rotational speed of 12,000 rpm. A) What is the speed of 12,000 rpm. A) What is the magnitude of the centripetal acceleration of magnitude of the centripetal acceleration of a red blood cell at a radial distance of 8.0 a red blood cell at a radial distance of 8.0 cm from the centrifuge’s axis of rotation? cm from the centrifuge’s axis of rotation? Compare this with g.Compare this with g.A ball attached to a string is swung with A ball attached to a string is swung with uniform motion in a horizontal circle over a uniform motion in a horizontal circle over a person’s head. If the string breaks, what person’s head. If the string breaks, what will be the trajectory of the ball?will be the trajectory of the ball?

More Examples…More Examples…

A car approaches a level circular curve with A car approaches a level circular curve with radius of 45.0 m. If the concrete pavement is radius of 45.0 m. If the concrete pavement is dry, what is the maximum (constant) speed at dry, what is the maximum (constant) speed at which the car can negotiate the curve? which the car can negotiate the curve? Two masses, mTwo masses, m11 = 2.5 kg and m = 2.5 kg and m22 = 3.5 kg are = 3.5 kg are connected by light strings and are in uniform connected by light strings and are in uniform circular motion on a horizontal frictionless circular motion on a horizontal frictionless surface, with rsurface, with r11 = 1.0m and r = 1.0m and r22 = 1.3m. The forces = 1.3m. The forces acting an the masses are Tacting an the masses are T11 and T and T22 respectively. respectively. Find the centripetal acceleration and the Find the centripetal acceleration and the tangential speed of mass 2 and mass 1.tangential speed of mass 2 and mass 1.

Angular AccelerationAngular Acceleration

When circular motion isn’t uniform, angular When circular motion isn’t uniform, angular velocity changes.velocity changes.

α , alpha, is used to denote angular acceleration. α , alpha, is used to denote angular acceleration. We consider only the case of constant angular We consider only the case of constant angular acceleration so acceleration so

ααavav = α = Δω/t = α = Δω/t

Then Δω = αt and ωThen Δω = αt and ωff = ω = ωii + αt + αt

Tangential AccelerationTangential Acceleration

If the rate of angular velocity changes, If the rate of angular velocity changes, then the object undergoes angular then the object undergoes angular acceleration and tangential acceleration:acceleration and tangential acceleration:

aatt = = Δv/t = Δ(rω)/t = rαΔv/t = Δ(rω)/t = rα

aat t = = rαrα

Linear and Rotational Linear and Rotational KinematicsKinematics

X = vX = vavav t t

VVavav = (v = (vff + v + vii)/2)/2

VVff = v = vii + at + at

XXff = x = xii + v + viit + ½ att + ½ at22

VVff22 = v = vii

22 + 2a + 2aΔΔxx

θ =θ = ω ωavavtt

ωωav av = (ω= (ωff+ω+ωii)/2)/2

ωωff = ω = ωii + αt + αt

θθff =θ =θii + ω + ωiit + ½ αtt + ½ αt22

ωωff22

= = ωωii2 2 + 2αΔθ+ 2αΔθ

ExamplesExamplesA CD accelerates uniformly from rest to its A CD accelerates uniformly from rest to its operational speed of 500 rpm in 3.0 sec. What is operational speed of 500 rpm in 3.0 sec. What is the angular acceleration of the CD during this the angular acceleration of the CD during this time? If the CD comes to a stop in 4.0 sec, what time? If the CD comes to a stop in 4.0 sec, what is the angular acceleration during that part of the is the angular acceleration during that part of the motion?motion?A microwave oven has a 30 cm rotating plate. A microwave oven has a 30 cm rotating plate. The plate accelerates from rest to a uniform rate The plate accelerates from rest to a uniform rate of 0.87 rad/s2 for 0.50 sec before reaching its of 0.87 rad/s2 for 0.50 sec before reaching its constant operational speed. How many constant operational speed. How many revolutions does the plate make before reaching revolutions does the plate make before reaching its operational speed? What are the operational its operational speed? What are the operational angular speed and tangential speed at its rim?angular speed and tangential speed at its rim?

Newton’s Law of GravitationNewton’s Law of Gravitation

Newton wondered whether the force Newton wondered whether the force pulling apples to the ground was the same pulling apples to the ground was the same force pulling the moon towards the Earth.force pulling the moon towards the Earth.

He recognized that, although the moon He recognized that, although the moon orbits the Earth in a circular path, that the orbits the Earth in a circular path, that the moon really is ‘falling’ towards the Earth moon really is ‘falling’ towards the Earth

Newton’s Cannon Thought Newton’s Cannon Thought ExperimentExperiment

Gravitational ForceGravitational Force

Gravitational Force Gravitational Force

Is always attractiveIs always attractiveIs proportional to the product of the Is proportional to the product of the massesmassesIs inversely proportional to the square of Is inversely proportional to the square of the distance between the massesthe distance between the massesG = 6.67 X 10 G = 6.67 X 10 -11-11 Nm Nm22/kg/kg22

ExamplesExamples

Compute the gravitational force between Compute the gravitational force between Earth and the Moon. MEarth and the Moon. Mmm = 7.4 x 10 = 7.4 x 102222 kg, kg, MME E = 6.0 X 10= 6.0 X 102424 kg, r kg, rEMEM = 3.8 X 10 = 3.8 X 1088 m. m.

Find the acceleration due to gravity acting Find the acceleration due to gravity acting on an object with mass Mon an object with mass Moo near the Earth’s near the Earth’s surface. rsurface. rEE = 6.4 X 10 = 6.4 X 1066mm

Satellite ExampleSatellite Example

Some communication and weather Some communication and weather satellites are launched into circular orbits satellites are launched into circular orbits above the Earth’s equator so they are above the Earth’s equator so they are synchronous with Earth’s rotation. At synchronous with Earth’s rotation. At what altitude are these satellites? what altitude are these satellites? RREE = 6.4 X 10 = 6.4 X 1066 km; km;

MMEE = 6.0 X 10 = 6.0 X 102424 kg, kg,

T = 365 daysT = 365 days