Physics a2 unit4_05_circular_motion_01 phyiscs circular motion
- 2. 1. To summarise the relationship between degrees and radians 2. To understand the term angular displacement 3. To define angular velocity 4. To connect angular velocity to the period and frequency of rotation 5. To connect angular velocity to linear speed
- 3. Angles can be measured in both degrees & radians : Arc length θ r The angle θ in radians is defined as the arc length / the radius For a whole circle, (360°) the arc length is the circumference, (2π r) ∴ 360° is 2π radians (or “rad”) Common values : 45° = π /4 radians 90° = π /2 radians 180° = π radians Note. In S.I. Units we use “rad” How many degrees is 1 radian?
- 4. Angular velocity, for circular motion, has counterparts which can be compared with linear speed s=Δx/Δt. Period of time (Δt) remains unchanged, but linear distance (Δx) is replaced with angular displacement Δθ measured in radians. Angular displacement Δθ r Δθ r Angular displacement is the number of radians moved
- 5. For a watch calculate the angular displacement in radians of the tip of the minute hand in 1. One second 2. One minute 3. One hour Each full rotation of the London eye takes 30 minutes. What is the angular displacement per second?
- 6. Consider an object moving along the arc of a circle from A to P at a constant linear speed for time Δt: Arc length Definition : The rate of change of angular displacement with time A “The angle, (in radians) an object rotates through per second” P r θ r ω = Δθ / Δt Where Δθ is the angle turned through in radians, (rad), yields units for ω of rads-1 This is all very comparable with linear speed, (or velocity) where we talk about distance/time
- 7. The period T of the rotational motion is the time taken for one complete revolution (2π radians). Substituting into : ω = Δθ / Δt ω = 2π / T ∴ T = 2π / ω From our earlier work on waves we know that the period (T) & frequency (f) are related T = 1/f ∴ f = ω / 2π
- 8. Considering the diagram below, we can see that the linear distance travelled is the arc length P Arc length r θ r A ∴ Linear speed (v) = arc length (AP) / Δt v = r Δθ / Δt Substituting... (ω = Δθ / Δt) v = ωr
- 9. A cyclist travels at a linear speed of 12 ms-1 on a bike with wheels which have a radius of 40 cm. The wheels rotate clockwise. Calculate: a. The frequency of rotation for the wheels b. The angular velocity for the wheels c. The angle the wheel turns through in 0.10 s in i. radians ii. degrees
- 10. The frequency of rotation for the wheels Circumference of the wheel is 2π r = 2π x 0.40m = 2.5m Time for one rotation, (the period) is found using s = Δd / Δt rearranged for Δt Δt = Δd / s = T = circumference / linear speed T = 2.5 / 12 = 0.21s f = 1 / T = 1 / 0.21 = 4.8Hz
- 11. The angular velocity for the wheels Using T = 2π / ω , rearranged for ω ω = 2π / T ω = 2π / 0.21 ω = 30 rads-1 Clockwise
- 12. The angle the wheel turns through in 0.10s in i radians ii degrees Using ω = Δθ / Δt re-arranged for Δθ Δθ = ω t Δθ = 30 x 0.10 Δθ = 3.0 rad = 3.0 x (360°/ 2π ) = 172° ≈ 1.7 x 102 °