I

NTRO TO

C

IRCULAR

‘Round and ‘round we go!

M

OTION

O BJECTIVES     Define & understand the following terms: period, frequency, angular displacement & angular velocity Understand and calculate centripetal force and acceleration Identify the forces providing the centripetal force (tension, friction, gravitational, electrical & magnetic) Qualitatively & quantitatively describe examples of circular motion including cases of vertical & horizontal circular motion

E NGAGE & E XPLORE   Hold the string and swing the ball around your head. Release when you believe the ball will hit the target. You have 3 opportunities to hit the target. Relay-race: get into teams of 3-4. When the first person hits the target, hand the ball off to the next team mate. The fastest group to get everyone to hit the target and sit down picks from the prize bucket.

A NALYSIS    When you hit the target, where was the ball in it’s circular path when you let it go?

What forces/properties of matter guided the ball’s motion once you let go?

What does this mean about the direction of the ball’s velocity when you let it go? Draw a picture to illustrate your idea.

 E XPLAIN : I MPORTANT NEW SYMBOLS & VOCABULARY Arc length – s – distance travelled in an arc      Radius – r- the distance from the center to the perimeter of the circular path Angle – θ – the angle subtended 𝜃 = 𝑠 𝑟 ; units: radians Angular velocity – ω – the rate of change of the angle 𝑣 𝜔 = = ∆𝜃 = 2𝜋𝑓 𝑟 ∆𝑡 Linear velocity – v – rate of displacement 𝑣 = ∆𝑥 ∆𝑡 Angular Acceleration – α – rate of change of the angular velocity 𝛼 = 𝑎 𝑡 𝑟 = ∆𝜔 ∆𝑡 = 𝜔𝑟

 E XPLAIN : I MPORTANT NEW SYMBOLS & VOCABULARY Frequency – f – number of complete revolutions per second; units: 𝑠 −1 = 𝐻𝑧  Period – T – the amount of time required for one complete revolution 𝑇 = 1 𝑓

E LABORATE Create 8 vocab tabs using the single sheet of paper as directed  Outside: symbol  Inside: definition, equation & picture  Use colors! This is homework if not finished by the end of class.

IB E XPLAIN : F RAME OF REFERENCE  Inertial frame of reference – what we’ve been exploring    Forces obey newton’s laws a frame of reference in which a body remains at rest or moves with constant linear velocity unless acted upon by net external forces Example: everything we’ve done so far  Rotating frame of reference (non-inertial)  If your frame of reference has a non-uniform, or accelerated motion, then the Law of Inertia will appear to be wrong, and you must be in a non-inertial frame of reference.

 Performing an experiment on an accelerating train

E LABORATE : EXAMPLES CALCULATIONS USING ANGULAR SPEED     Calculate the angular speed of the second hand on a clock.

Calculate the angular speed of the minute hand on a clock.

A car drives round a circular track of radius 1.0 km at a constant speed of 26 ms -1 . What is the angular speed?

A disc rotates at an angular speed of 4.7 rad s center, what is it’s linear speed?

-1 . An object is placed a distance of 4.0 cm from the

E VALUATE : Y OU TRY !

    Calculate the angular speed of a 4000 rpm (rotations per minute) CD Rom drive.

Calculate the angular speed of the hour hand on a clock.

A disc rotates at an angular speed of 4.7 rad s -1 . An object is place a distance of 8.0 cm from the center, what is it’s linear speed? Compare the answer to the last problems with that of the last example. What happens to the linear speed of an object as it moves outward at the same angular velocity? Why do you think this is?

E XPLAIN : W HAT MAKES THINGS MOVE IN CIRCLES ?

Video: Circular Motion  Velocity is speed in a direction   In circular motion, direction is changing (even if speed is constant), so the velocity is changing which means that the object is accelerating –

Centripetal acceleration a c

 𝑎 𝑐 = 𝑣 2 𝑟 = 4𝜋 2 𝑟 𝑇 2 Accelerations are caused by net forces, so there must be a net force acting on the object –

Centripetal force F c

 Directed inward – center-seeking  𝐹 𝑐 = 𝑚𝑣 2 𝑟 = 𝑚𝜔 2 𝑟

E LABORATE : E XAMPLES

E VALUATE : Y OU TRY !

 The speeds of a 600-kg roller coaster car at the top of three consecutive hills are shown below. The radii of the hills are shown. Determine the acceleration of and net force and normal force experienced by the car at the top of each hill.

E XPLAIN MOTION – E QUATIONS OF ROTATIONAL We can use rotational analogs of the linear equations we already know to solve for unknown quantities, just like we did before.

Equations for constant angular acceleration



𝜃 = 𝜔

𝑎𝑣𝑔

𝑡

Equations for constant linear acceleration



𝑥 = 𝑣

𝑎𝑣𝑔

𝑡



𝜔 = 𝜔

0

+ 𝛼𝑡



𝑣 = 𝑣

0

+ 𝑎𝑡



∆𝜃 = 𝜔

0

𝑡 +



𝜔

2 1 2

𝛼𝑡

2 

∆𝑥 = 𝑣

0

𝑡 +

1 2

𝑎𝑡

2

= 𝜔

0 2

+ 2𝛼∆𝜃



𝑣

2

= 𝑣

0 2

+ 2𝑎∆𝑥 Can you see the similarities?

E XTEND    International mindedness: International collaboration is needed in establishing effective rocket launch sites to benefit space programs.

TOK: Foucault’s Pendulum – simple, observable proof of the Earth’s rotation Use: Playground/amusement park rides

P RACTICE  Try to get 100 points on the kinetic books assignment. You may work with a partner.