A –Level Physics: Further Mechanics- Angular Velocity and Acceleration

Objectives: Spec point: 20. know and understand Newton’s third law of motion and know the properties of pairs of forces in an interaction between two bodies Spec point: 21 understand that momentum is defined as p = mv Spec point: 22 know the principle of conservation of linear momentum, understand how to relate this to Newton’s laws of motion and understand how to apply this to problems in one dimension Additional Skills Gained: Estimation Laying out answers Derivation

Starter Activity FLASHBACK: Define all three of Newton’s Law of Motion and State the requirements of a Newton Pair

Circular Motion Angular Displacement is “the angle through which a point has been rotated in a given direction”. As it is an angle it can be expressed in both radians and degrees. If you are going ANTICLOCKWISE then you get a positive displacement (and vice versa for clockwise). Express the following quantities in the opposite form (e.g. Degrees to Rads) 1) 45° 2) 360° 3) π rad 4) π 8 rads 5) 11.25° 6) 2.2 rads Inelastic- gradually reduces in vertical displacement and distance so energy has been transferred into another form (lost) therefore kinetic is NOT conserved. NB: To convert from rads to degrees you multiply by 180 π and to do the reverse you multiply by π 180

What might “linear speed” mean? Angular Velocity Angular Velocity is simply “the angle an object rotates through per second”. It’s the angular version of velocity=displacement/time! The unit for angular velocity is (as you could expect)…rad s-1 . The symbol for angular velocity is lower case omega (ω). The angle MUST be in radians before calculating The angular velocity (ω) and the linear speed (v) are also linked. They are linked by the radius of the circle! v= ωr Linear Speed= Tangential Speed (the speed at which the object would travel if the centripetal force were removed) What might “linear speed” mean?

Calculation Practice: Simple Calculate the angular velocity of: the London Eye which rotates once every half an hour A CD which when first switched on rotates at 200rpm (rotations per minute) ANSWERS: Calculate the angular velocity of: the London Eye which rotates once every half an hour. 30mins= 1800s for one rotation. So as 2π is a whole rotation, w=2π/1800= 3.5x10-3rads-1 A CD which when first switched on rotates at 200rpm (rotations per minute) In the same way, 200rpm is 3.3rps, so 3.3 rotations= 6.6 π which divide by 1(second)= 6.6 π = 20.7rad s-1

Calculation Practice: Synoptic a) The information on a DVD is recorded as a series of bumps in the tracks on the disc. The information is then recorded by the laser beam. Calculate the speed of the bumps moving past the laser head at 4.0cm from the centre of the disc which is rotating at 430rpm (4 marks)” b) Explain in terms of wave interference how the bumps and gaps in the disc correspond to detectable data when they pass under the laser head (4 marks)

Calculation Practice: Synoptic a) The information on a DVD is recorded as a series of bumps in the tracks on the disc. The information is then recorded by the laser beam. Calculate the speed of the bumps moving past the laser head at 4.0cm from the centre of the disc which is rotating at 430rpm (4 marks)” The DVD is moving at 430rpm which is 7.16rps. This means that the angle it has rotated through in one second is 14.32π. This works out as 45rad s-1. As v=wr, and r=0.04m, the linear speed is 1.8ms-1. b) Explain in terms of wave interference how the bumps and gaps in the disc correspond to detectable data when they pass under the laser head (4 marks) The dips/gaps in the CD are equal to half a wavelength. Therefore when light passes over the CD, at times ALL the light hits either a dip or a bump. In this instant all the light is reflected in phase with no/whole wavelength path difference and so constructively interferes indicating a ‘1’. Whereas if the beam is partly on a gap/bump then some light travels half a wavelength more/less (path difference) therefore resulting in destructive interference and as such a ‘0’

Cyclotrons Use the diagrams above to help you RECALL and EXPLAIN how a cyclotron accelerates particles! (4 marks)

Real life context- Cyclotrons Cyclotrons accelerate particles to very high linear speeds by alternating the polarity of dees that are within a magnetic field. The particles start at a central point and then spiral outwards. The angular velocity of the particles always remain constant. If the angular velocity is constant, then why is the linear speed changing? PRACTICE QUESTION: Assume that a particle of mass 1.45x10-18kg had been injected into the centre of a cyclotron. When in motion the particle rotates through 360˚ in 35microseconds. Calculate the linear speed of a particle when it is 1.5m from the centre of rotation Linear speed (v) = wr. So despite the fact that the angular velocity is constant, the radius of the particle’s circular motion is increasing thus increasing v. As angular velocity= angle over time, this is 2pi /35x10-6 = 1.795 x 10^5 rads-1. Put this in the other equation (v=wr) you then multiply by 1.5 to get 2.7x10^5 (2 s.f)

Frequency and Time Period You should remember from your work on oscillations that the frequency is the complete number of revolutions (or oscillations for waves!) per second! As such, the period (T) is the time taken for one complete revolution in seconds. So we get 𝑓= 1 𝑇 . For a complete rotation an object will go through 2π radians so: ω= 2 πf OR T= 2π ω Earth takes 365.25 days which = 8766 hours= 3.16x10^7s. 2pi/ANS= 2x10^-7 rad s-1 Calculate the angular velocity of the Earth! (1 year=365.25 days)

Instantaneous Velocity Even if an object is orbiting at a constant linear speed, its velocity must be changing as it is changing direction. If it is changing velocity then it must be accelerating. To discover the value to this centripetal (central finding) acceleration we must discover HOW QUICKLY the direction is changing). Throughout the next steps; draw the diagram with accuracy…. (see page 97 in revision guide or L3 in SOW).

Research Use pages 26-31 in the text books and pages 25-27 (worksheet) to write notes on centripetal force. Include any worked examples and then come up with explanation as to why a water skier produces a huge plume of water directed to the outside of a curved path. (see page 97 in revision guide or L3 in SOW).

Independent Study Complete questions on pages 35-37 AND revise all of the content covered so far this term(further mechanics)for an exam on Friday. This will include: collisions, impulse, circular motion and centripetal force. (see page 97 in revision guide or L3 in SOW).