PHY 2311

PHYSICS 231Lecture 16: Centripetal acceleration

Remco ZegersWalk-in hour: Tue 4-5 pm

Helproom

PHY 2312

Last lecture...

t

ttt

t

if

if

0lim

Average angularvelocity (rad/s)

Instantaneous Angular velocity

t

ttt

t

if

if

0lim

Average angularacceleration (rad/s2)

Instantaneous angularacceleration

2 rad 3600

10 2/360 rad1 rad 360/2 deg

PHY 2313

And...

r

v

Angular and linear velocity are related:

r

a

Angular and linear acceleration arerelated:

demo: tangential velocity

PHY 2314

Rotational motion

Angular motion

(t)= (0)+(0)t+½t2

(t)= (0)+t

PHY 2315

gears

r1=0.3 r2=0.15

r3=0.7 m

If 1=3 rad/s, how fastIs the bike going? b) What if r1=0.1 m ?

V1= 1r1=3*0.3=0.9 m/sV1=V2 (because of the chain)2=V2/r2=0.9/0.15=6 rad/s3=2 (connected)V3=3r3=6*0.7=4.2 m/s

b)

V1= 1r1=3*0.1=0.3 m/sV1=V2 (because of the chain)2=V2/r2=0.3/0.15=2 rad/s3=2 (connected)V3=3r3=2*0.7=1.4 m/s

PHY 2316

quiz (extra credit)

An object is moving in a circle with radius R=2 m. If theradius is doubled and the angular velocity remains the same,the linear velocity will:

a) halfb) remain the samec) doubled) quadruple e) whatever

PHY 2317

for old times sake…

A man is standing on a platform resting on a frictionlesssurface. At the end of the platform a wall is placed andfixed to the platform.The man throws a ball against the wall and it bounces back over the man as shown in the figure. After the ball has bounced against the wall…

a) the platform moves to the leftb) the platform moves to the rightc) the platform remains at its place

PHY 2318

Driving a car through a bend

Is there a force that pushesyou away from the center ofthe circle?

•Newton’s first law: If no net force is acting on an object, it will continue with the same velocity (inertia of mass)

•Velocity is a vector (points to a direction)•If no net force is acting on an object, it will not change its direction.

•A force is acting on the car (steering+friction) but you tend to go in the same direction as you were going!

•It is not a force that pushes you, but the lack of it!•The side door will keep you from falling out: it exerts a force on you and you exert a force on the door (F21=-F12)

PHY 2319

Centripetal acceleration

t

v

tt

vva

if

if

The change in velocity is not the change in speedbut in direction.

Sin(/2)=(s/2)/r Sin(/2)=(v/2)/v

v=s*(v/r)t t

(s/2)/r=(v/2)/v

vs/t

ac=v2/r

PHY 23110

Centripetal acceleration

ac=v2/r directed to the center of the circular motion

Also v=r, so ac=2r

This acceleration can be caused by various forces:• gravity (objects attracted by earth) • tension (object making circular motion on a rope)• friction (car driving through a curve)• etcThis acceleration is NOT caused by a mysterious force

PHY 23111

Forces that can cause centripetal acceleration.

Object swinging on a rope.

T

T

F=ma

T=mac

T=mv2/r=m2r

An object with m=1 kg is swung with a rope of length 3 m around with angular velocity =2 rad/s. What is the tension in the rope?

T=m2r=1*22*3=12 N

PHY 23112

Lifting by swinging

Swinging mass (m1) with velocity v

Hanging mass (m2)

r

What is the relation between v and rthat will keep m2 stationary?

T

T

Fg=m2g

m1: T=m1am2: -T=m2g

v out of paper

a=-m2g/m1

Also: ac=(-)v2/r v=linear/tangentialvelocity of m1 v2/r= m2g/m1

If m1 slows down, r must go downso m2 sinks.

demo

PHY 23113

Swinging mass (m1) with velocity v

Hanging mass (m2)

r

T

T

Fg=m2g

v out of paper

v2/r= m2g/m1

question

If the velocity of m1 is increasedto twice the original value,the radius of orbit a) decreases to ¼ of its original valueb) decreases to ½ of its original valuec) remains the samed) increases to 2x its original valuee) increases to 4x its original value

PHY 23114

A car going through a bendA car is passing through a bend withradius 100 m. The kinetic coefficientof friction of the tires on the road is0.5. What is the maximum velocity thecar can have without flying out of thebend?

F=makn=mac

kmg=mv2/r0.5*9.81=v2/100v=22 m/s

PHY 23115

2001: A space odysseyA space ship rotates witha linear velocity of 50 m/s.What should the distancefrom the central axis tothe crew’s cabin’s be so thatthe crew feels like they areon earth? (the floor of thecabins is the inside of theouter edge of the spaceship)

F=ma mg=mv2/r so r=v2/9.8 and thus r=255 m

The rotating spaceship has an acceleration directed towardsthe center of the ship: the ‘lack’ of forces acting on the crewpushes them against the ship.

PHY 23116

Conical motion

What is the centripetal acceleration ifthe mass is 1 kg and =20o?

mg

TTcos

Tsin

Vertical direction:F=maTcos-mg=0So T=mg/cos

Horizontal direction:F=mac

Tsin=mac

mgsin/cos=mgtan=mac

ac=gtan=9.8*0.36=3.6 m/s2

demo

PHY 23117

A general strategy

•As usual, make a drawing of the problem, if not given.

•Draw all the forces that are acting on the object(s) under investigation.

•Decompose each of these into directions toward the center of the circular path and perpendicular to it.

•Realize that Fto center=mac=mv2/r

PHY 23118

question

A point on the rim of a 0.25 m radius rotating wheel hasa centripetal acceleration of 4.0 m/s2. What is the linearspeed of the wheel? (Fto center=mac=mv2/r)a) 1.0 m/sb) 2.0 m/sc) 3.2 m/sd) 4.0 m/se) 8.0 m/s

ac=v2/r 4=v2/0.25 v2=16 v=4 m/s