Circular motion. The track cyclist leans in The road banks

Circular motion

The track cyclist leans in

The road banks

A turning airplane ‘banks’

Race drivers learn how to

corner

Staying on the road on a curve is the

mark of a good driver

Oops!

It all starts with the vector description of

position

And my Laws!

An object at position r1 moves to position r2 in time t

r1

r2v1

v2

Motion is at constant speed; |v1| = |v2|

Look at the changing velocity vectors

The velocity vectors v are everywhere perpendicular to radius r.

r1

r2v1

v2

The triangle formed by r1 and r2 defines the vector r

v1

v2

vr2

r1

r

The triangle formed by vectors v1 and v2 is similar because the r and v vectors are mutually perpendicular.

The v triangle defines the vector V

If there’s v, there must be acceleration

Or else its goodbye, satellite! a

The vector v points radially inwards

Thus the acceleration vector a = v/t must also be radially inwards.

This is called centripetal (center-seeking) acceleration.

v1

v2

vr2

r1

r

By similar triangles,

r v

r vr

v vr

v1

v2

vr2

r1

r

Divide both sides by t:

2

rv v

r

v v r v

t r t r

This is the magnitude of centripetal acceleration.

v1

v2

vr2

r1

r

Centripetal acceleration keeps you moving in a circle

Without that acceleration, motion

continues in a straight line.

There’s a neat way to derive this using Calculus, but we’ll leave

that as a challenge for you

r1

r2v1

v2

2

2

cos ,sinr r

drv

dt

dv d ra

dt dt

Of use is the radial unit vector:

r1

r2v1

v2

2

2

ˆ ˆˆ cos ,sin cos sin

ˆ| | . Since |r| is constant,

ˆ ˆ| | | |

r i j

r r r

dr dv d rv r a r

dt dt dt

And the quantity known as angular velocity:

r1

r2v1

v2

,ds d

s r v rdt dt

d v

dt r

Angular velocity is measured in radians per second.

Angular velocity?2

, (it takes T sec to make one full circle)d v

dt r T

There are two kinds of circular motion:

Uniform Circular Motion

Angular velocity (and therefore speed) are constant. The

centripetal acceleration vector is directed at right angles

to the velocity vector.

Non-uniform circular motion

Tangential acceleration changes the angular velocity and therefore the speed of the bug.

Radial acceleration only changes the direction.

Non-uniform circular motion

Procedure: determine |at| from the change in speed. Determine |ar|. The vector acceleration a has

2 2

1

| |

tan

t r

t

r

a a a

a

a

ProblemsAn airplane goes into a circular dive at a speed of 550 km/hr. The pilot experiences “2.5 g’s” (pilot lingo for ‘an acceleration equal to 2.5 times earth gravity’).

What must be the radius of the dive?

ProblemsIn science fiction movies, a space station rotates to provide ‘artificial gravity.’ Suppose a station that will be 300 m in radius is to rotate fast enough to provide at least ½ earth gravity for the comfort of its occupants.

What should be the station’s minimum rotational speed, expressed in revolutions per minute?If you stand at the outer edge of the station and I’m at the hub, how fast do I think you are going (what is your speed in m/s)?

ProblemsA loop-de-loop ride at an amusement park traverses a circular arc of radius 24 m. In order to keep the cars on the track when at the top of the loop, what must be theminimum speed of the ride?

Magnitude of centripetal acceleration

a = v2/r

Since velocity is only tangential, acceleration can only be radial andtherefore acceleration only changes the direction of the velocity.

Problems

Going around a corner, a driver enters a curve of radius 150 m while traveling at 27 m/s.Realizing he is going too fast, he slows to 24 m/s in 4 seconds.

What is the average acceleration experienced by the driver while slowing?