Phy 201: General Physics I

Chapter 5: Uniform Circular Motion

Lecture Notes

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:

ˆ ˆyxc x y

vva =a a x + y

t t

vr

2

c c

va =a =

r

v

Derivation of Centrifugal AccelerationConsider 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:

x

v

r

y

x

y

The direction of position and velocity vectors both shift by the same angle, resulting in the following relation:

2

c

r v v r v v r= v= =

v a =

r v r r t rt

r1

r2

r

Position only

v1v2

v

Velocity onlyr

2

v2

time, t2

r1

v1

time, t1

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

c cF = ma

2

c c c c

mvF = F = ma F = F = ma =

r

2

c c c

mvF = F = ma =

r

Centripetal Force & GravitationAssume the Earth travels about the Sun in UCM:

Questions:1. What force is responsible for maintain the Earth’s circular orbit?

2. Draw a Free-Body Diagram for the Earth in orbit.

3. Calculate gravitational force exerted on the Earth due to the Sun.

4. What is the centripetal acceleration of the Earth?

5. What is the Earth’s tangential velocity in its orbit around the Sun?

Sun

Earth

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 2 radians (360o) in a complete rotation/revolution• Since v = r (rotational speed x radius), the artificial gravity is:

• Thus, the (artificial) gravitational strength is related to size of the structure (r), rotational rate () and square of the period of rotation (T2)

2

c artificial

va = g =

r

{where T is the period of the motion}2

= = T t

22

artificial 2

4 rg = ω r =

T

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:

• Can you think of other objects that undergo similar motions?