Centripetal Centripetal Acceleration/ForceAcceleration/Force

Centripetal Force/Acceleration Centripetal Force/Acceleration DefinitionDefinition

Centripetal force:Centripetal force: Any force that causes curved path motionAny force that causes curved path motion Measured in Measured in NewtonsNewtons provided that provided that mass in kgmass in kg and and acceleration acceleration

is in m/sis in m/s22

Examples:Examples: Gravity causing a satellite to orbit a planetGravity causing a satellite to orbit a planet

Friction causing a vehicle to round a curveFriction causing a vehicle to round a curve A string-tethered mass being whirled in a circleA string-tethered mass being whirled in a circle

Centripetal acceleration:Centripetal acceleration: The resulting acceleration from a centripetal forceThe resulting acceleration from a centripetal force The rate change in velocity The rate change in velocity directiondirection. . Measured in m/sMeasured in m/s22 in SI units in SI units

• The word centripetal means “center-seeking”.

Centripetal Acceleration/Force Centripetal Acceleration/Force Equations DerivationEquations Derivation

v1 v1 v1 v1

v4 v3

v2v

v# is the tangential velocity of the object at various times.

Δv=v4-v1 Δv=v2-v1Δv=v3-v1

Δv=v-v1

v is an infinitesimal time greater than v1

Notice the direction of Δv direction as the time interval decreases

θ

θθ

θθθ

θθ

Direction of

motion The speed of the object is uniform

Centripetal Acceleration Centripetal Acceleration DerivationDerivation

(continued)(continued)

r

tv

v

v

r

v

t

v 2

Δv

magnified

r

r

s=d=vΔt

Along thearc 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 travelsalong the curved path.

Δv is the velocity directionchange.

r

va

2

c

r

va

2

c

s

Centripetal Acceleration/Force Centripetal Acceleration/Force EquationsEquations

r

va

2

c

ac = centripetal acceleration (rate change in velocity direction)v = speed of objectr = radius of pathω = angular speed f = frequencyT=period

Fc= mac Fc= centripetal force

v=rω ac=(rω)2/r

ac=ω2r

ω = 2πf ac = (2πf)2r

ac=4π2rf2

f=1/T

2

2

c T

r4a

Centripetal Force/Acceleration/Tangential Velocity Centripetal Force/Acceleration/Tangential Velocity OrientationsOrientations

Fc

Fc

Fc

Fc

Fc

vv

v

v

v

Fc =centripetal forceac = centripetal accelerationv= tangential velocity

ac

ac

ac

The centripetal force and acceleration are always directed towards the center of the path.

For uniform circular motion to occur thecentripetal force must be perpendicular to the tangential velocity at all times.

Direction of

motion

ac

ac

Removal of Centripetal Removal of Centripetal ForceForce

If the centripetal force isremoved, then the masstravels in a straight line, tangent to it current path. (Newton’s 1st 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”.

““Centrifugal Force” Centrifugal Force” AnimationAnimation

How Force Direction Influences How Force Direction Influences Centripetal Acceleration and Linear Centripetal Acceleration and Linear

AccelerationAcceleration

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 Orientation Resulting Motion