Phy 201: General Physics I

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# Phy 201: General Physics I - PowerPoint PPT Presentation

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

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### Phy 201: General Physics I

Chapter 5: Uniform Circular Motion

Lecture Notes

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:

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:

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

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?
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…

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)
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!
Vertical Circular Motion
• Can you think of other objects that undergo similar motions?