1. Centripetal Acceleration/Force

2. Centripetal Force/Acceleration Definition Centripetal force: Centripetal force: Any force that causes curved path motion Any force that causes curved path motion Measured in Newtons provided that mass in kg and acceleration is in m/s 2 Measured in Newtons provided that mass in kg and acceleration is in m/s 2Examples: Gravity causing a satellite to orbit a planet Gravity causing a satellite to orbit a planet Friction causing a vehicle to round a curve Friction causing a vehicle to round a curve A string-tethered mass being whirled in a circle A string-tethered mass being whirled in a circle Centripetal acceleration: The resulting acceleration from a centripetal force The resulting acceleration from a centripetal force The rate change in velocity direction. The rate change in velocity direction. Measured in m/s 2 in SI units Measured in m/s 2 in SI units The word centripetal means “center-seeking”.

3. Centripetal Acceleration/Force Equations Derivation v1v1 v1v1 v1v1 v1v1 v4v4 v3v3 v2v2 v v # is the tangential velocity of the object at various times. Δv=v 4 -v 1 Δv=v 2 -v 1 Δv=v 3 -v 1 Δv=v-v 1 v is an infinitesimal time greater than v 1 Notice the direction of Δv direction as the time interval decreases θ θ θ θ θ θ θθ Direction of motion The speed of the object is uniform

4. Centripetal Acceleration Derivation (continued) ΔvΔv magnified r r s=d=vΔt Along the arc length The triangles are similar as established on the previous slide. v v v is the velocity magnitude (speed) Which is the same as the object travels along the curved path. Δv is the velocity direction change. s

5. Centripetal Acceleration/Force Equations a c = centripetal acceleration (rate change in velocity direction) v = speed of object r = radius of path ω = angular speed f = frequency T=period F c = ma c F c = centripetal force v=rωa c =(rω) 2 /r ac=ω2rac=ω2r ω = 2πf a c = (2πf) 2 r a c =4π 2 rf 2 f=1/T

6. Centripetal Force/Acceleration/Tangential Velocity Orientations FcFc FcFc FcFc FcFc FcFc v v v v v F c =centripetal force a c = centripetal acceleration v= tangential velocity acac acac acac The centripetal force and acceleration are always directed towards the center of the path. For uniform circular motion to occur the centripetal force must be perpendicular to the tangential velocity at all times. Direction of motion acac acac

7. Removal of Centripetal Force If the centripetal force is removed, then the mass travels in a straight line, tangent to it current path. (Newton’s 1 st law, Inertia) v v There is no force that causes the object to move outward in a straight line once the centripetal force is removed. The inertia of the mass keeps it moving in a straight line at constant speed. This fictitious force is called a centrifugal force. Centrifugal force actually describes the resulting motion due to the absence of a centripetal force. The word centrifugal means “center-fleeing”.

8. “Centrifugal Force” Animation

9. How Force Direction Influences Centripetal Acceleration and Linear Acceleration v F Increase in speed (Linear acceleration) Straight line motion No change in direction v F Change in direction (centripetal acceleration) If the force and velocity are always perpendicular then circular motion will result. No change in speed v F Change in speed and direction Increase or decrease in speed depending on the force direction. A spiral path inward or outward depending on the force direction Velocity/Force OrientationResulting Motion