3.1 Motion in a Circle Gravity and Motion The force that keeps a body firmly attached to the ground is the same force that keeps the Moon in orbit around the Earth and keeps the Earth going around the Sun and is called gravitational force. All satellites and planets orbit larger bodies in elliptical orbits. These orbits are very close to being circles, and therefore circular orbits will be studied in detail in this chapter. Uniform Circular Motion Any object that is travelling in a circle at a constant speed is said to be in uniform circular motion. Acceleration is the change in velocity which is a vector and has both speed and direction. An object undergoing uniform circular motion will be accelerating, even though it has a constant speed, due to its changing direction.

3.1 Motion in a Circle Identifying Centripetal Acceleration vo vo Δv vf -Δv Δt Using just the velocity vectors, the change in velocity shows the direction of the acceleration. vf As the change in time approaches zero, Δt  0, the direction of the change in velocity becomes ever closer to the center of the circular path. An object moving in uniform circular motion. The velocity of the object at two different positions of the object at two different times are shown above. The direction of the acceleration of an object moving in uniform circular motion is towards the center of the circle and is called centripetal acceleration

3.1 Motion in a Circle Direction of Centripetal Acceleration vo ΔR vo ѳ R0 ѳ R0 v1 -Δv R1 ѳ R1 Similar Triangles v1 Radius Vectors: Velocity Vectors: Examine two triangles formed when a body moves through uniform circular motion. v1 + (-v0) = Δv R1 + (-R0) = ΔR R0 + ΔR = R1 (-v0) + Δv = v1 At very small time intervals R1 = R0 = R and v1 = vo = v and: Δv v = ΔR R

3.1 Motion in a Circle Defining Centripetal Acceleration Δv v Δv v R = From previous slide: = Therefore: ΔR R Δv v ΔR And: a = = Δt R Δt R0 As Δt, time interval, becomes smaller, Δt  0, R0 = R1 and ΔR = Δs (arc length). Δs ΔR ѳ ΔR Δs v = = Δt Δt R1 ΔR = v * Δt Finally: v * v * Δt v * v Centripetal Acceleration Equation a = = ac = v2 R R Δt R (Magnitude of acceleration)

3.1 Motion in a Circle Another Way to Calculate Centripetal Acceleration Distance around a circle is called circumference and is equal to C = 2πR. R d 2πR v = = Δt Δt v2 (2πR)2 4π2R2 And: ac = = = R R * Δt2 R * Δt2 4π2R Centripetal Acceleration ac = (where Δt = T , the period) T2

3.1 Motion in a Circle Centripetal Force Remember Newton’s 2nd Law: Fnet = m * a Therefore centripetal Force (center-seeking force) must be equal to: Fc = m v2 R Fc = m * 4π2R T2 or

3.1 Motion in a Circle In this section, you should understand how to solve the following key questions. Page#158 Quick Check #1 – 3 Page #159 Practice Problems 3.1.1 Centripetal Force #1– 3 Page #160– 162 3.1 Review Questions #1 – 12