1. L12-s1,7 Physics 114 – Lecture 12 Chapter 5 Circular Motion; Gravitation We have so far considered motion along a straight line – F net acts along line of motion or is 0. What happens if F net acts at an angle to line of motion? → motion follows a curve, c.f. planetary orbits First we discuss how an angle may be measured in radians

2. L12-s2,7 Physics 114 – Lecture 12 The angle, θ, measured in radians, is given by, θ = s/r, where s is measured along the arc. This value is independent of the value of the radius Since the circumference of a circle, c = 2πr → 1 revolution = 360 0 = 2π radians = 2π c This is a very natural way to measure an angle. Notice that an angle does not have physical units. θ r s

3. L12-s3,7 Physics 114 – Lecture 12 §5.1 Kinematics of Uniform Circular Motion Uniform circular motion → Motion around a circle at constant speed. Velocity is not constant in this case since the direction is continually changing Lets analyze this motion Two parameters  v, r with physical units L/T and L Units of a  L/T 2 – only way to accomplish this is a = v 2 /r X constant Direction: natural axes along normal, along tangent and along radius  can only be along radius

4. L12-s4,7 Physics 114 – Lecture 12 Δθ r r → Mag of = v Definition of angle → Δθ = Δv/v, where Δv is mag of A B Definition of angle → Δθ = (arc AB)/r → arc AB = r Δθ

5. L12-s5,7 Physics 114 – Lecture 12 Speed = const = v = (Dist trav)/Time elapsed v = (Arc AB)/Δt = r Δθ/Δt → Δθ/Δt = v/r Δθ = Δv/v → Δv = vΔθ → a = Δv/Δt = vΔθ/Δt = v X v/r = v 2 /r Direction of ? Note that as Δt → 0, Δθ → 0 As Δθ → 0, the angle between, and hence, and approaches 90 0 → is directed inwards along the radius

6. L12-s6,7 Physics 114 – Lecture 12 Summarize: for motion at constant speed, v, in a circle of radius,r, the acceleration is: a R = v 2 /r directed radially inwards towards the center Period of Motion: Time, T, taken for one revolution Frequency of Motion: Number of revolutions per unit of time, usually in 1s. 1rev/s ≡ 1 hertz (hz) Note that fT = 1 → f = 1/T and T = 1/f Orbital speed, v = distance traveled/time elapsed → v = distance traveled in 1 revolution/time elapsed → v = 2πr/T

7. L12-s7,7 Physics 114 – Lecture 12 §5.2 Dynamics of Circular Motion Using Newton’s 2 nd Law: ΣF R = F T = ma R = mv 2 /r Lets see some examples §5.3 Highway Curves, Banked and Unbanked Discuss r