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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

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phy 201 general physics i

Phy 201: General Physics I

Chapter 5: Uniform Circular Motion

Lecture Notes

uniform circular motion

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:
derivation of centrifugal 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
  • 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
centripetal force gravitation

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
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
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
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
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?