Phy 201: General Physics I

1. Phy 201: General Physics I Chapter 5: Uniform Circular Motion Lecture Notes

2. r v Uniform Circular Motion UCM, a special case of 2-D motion where: • The radius of motion (r) is constant • The speed (v) is constant • The velocity changes as object continually changes direction • Since is changing (v is not but direction is), the object must be accelerated: • The direction of the acceleration vector is always toward the center of the circular motion and is referred to as the centripetal acceleration • The magnitude of centripetal acceleration:

3. y v r y q x x v2 time, t2 time, t1 v1 r2 Dr r1 r1 Position only q Velocity only r2 Dv v1 q v2 Derivation of Centrifugal Acceleration Consider an object in uniform circular motion, where: r = constant {radius of travel} v = constant {speed of object} The magnitude of the centripetal acceleration can be obtained by comparing the similar triangles for position and velocity: The direction of position and velocity vectors both shift by the same angle, resulting in the following relation:

4. Centripetal Force • Centripetal force is the center-seeking force that pulls an object into circular motion (object is not in equilibrium) • The direction of the centripetal force is the same as the centripetal acceleration • The magnitude of centripetal force is given by: • For uniform circular motion, the centripetal force is the net force: Notes: • the faster the speed the greater the centripetal force • the greater the radius of motion the smaller the force

5. Sun Earth Centripetal Force & Gravitation Assume the Earth travels about the Sun in UCM: Questions: • What force is responsible for maintain the Earth’s circular orbit? • Draw a Free-Body Diagram for the Earth in orbit. • Calculate gravitational force exerted on the Earth due to the Sun. • What is the centripetal acceleration of the Earth? • What is the Earth’s tangential velocity in its orbit around the Sun?

6. Centripetal Force vs. Centrifugal Force • An object moving in uniform circular motion experiences an outward “force” called centrifugal force (due to the object’s inertia) • However, to refer to this as a force is technically incorrect. More accurately, centrifugal force _______. • is a consequence of the observer located in an accelerated (revolving) reference frame and is therefore a fictitious force (i.e. it is not really a force it is only perceived as one) • is the perceived response of the object’s inertia resisting the circular motion (& its rotating environment) • has a magnitude equal to the centripetal force acting on the body Centripetal force & centrifugal force are NOT action-reaction pairs…

7. Artificial Gravity • Objects in a rotating environment “feel” an outward centripetal force (similar to gravity) • What the object really experiences is the normal force of the support surface that provides centripetal force • The magnitude of “effective” gravitational acceleration is given by: • If a complete rotation occurs in a time, T, then the rotational (or angular) speed is • Note: there are 2p radians (360o) in a complete rotation/revolution • Since v = wr(rotational speed x radius), the artificial gravity is: • Thus, the (artificial) gravitational strength is related to size of the structure (r), rotational rate (w) and square of the period of rotation (T2)

8. Banked Curves • Engineers design banked curves to increase the speed certain corners can be navigated • Banking corners decreases the dependence of static friction (between tires & road) necessary for turning the vehicle into a corner. • Banked corners can improve safety particularly in inclement weather (icy or wet roads) where static friction can be compromised • How does banking work? • The normal force (FN) of the bank on the car assists it around the corner!

9. Consider the forces acting on a motorcycle performing a loop-to-loop: Vertical Circular Motion • Can you think of other objects that undergo similar motions?