October 2014

Uniform Circular Motion

Uniform circular motion

What does the word “uniform” mean here? constant radius and constant speed

Velocity vector is tangent to the path at each instant, so direction of velocity vector changes all the time as the object moves in circle.

Therefore, there must be an inward-directed acceleration that causes the velocity to change its direction.

Circular Motion is towards the center.

is tangential to the motion

Speed is constant, changes

force directed towards the center is what causes the acceleration (e.g. gravity,

a string)

If the force is removed, the ball will continue in a

straight line at the speed it had.

Check your understanding …

When a wheel rotates about a fixed axis, do all the points on the wheel have the same tangential speed?

Yes!

Do they all have the same velocity?

No!

Check your understanding …

Circular Motion EquationsVt= 2πr/T

ac = vt2/r

Where Vt = tangental velocity

R = radius T = period (time required to make one complete circle)

ac = centripetal acceleration

Circular Motion– We do

The radius of a spacecraft orbiting earth is 6.67 x 106 m. If it orbits

earth in 5292 seconds, what is the velocity of the spacecraft?

Vt= 2πr/T = 2*π*6.67X106m / 5292 sec = 7920 m/s

Circular Motion– We do

Jimmie Johnson is driving his #48 Lowe’s NASCAR around a bend that has a radius of 70 meters. It takes him 30 seconds to travel the track.

What was the centripetal acceleration of Jimmie John’s #48

Lowe’s NASCAR?

Strategy: find vt, then find ac.

Vt= 2πr/T = 2π*70m/60sec = 7.32m/s

ac = vt2/r = (7.32 m/s)2 / 70 m = 0.8 m/s2

NOTICE! T is time to go around a full circle. Jimmie went ‘around a bend’ which is half a circle. Thus, we need to double his time to get T.

Centripetal Acceleration – You doa. A girl sits on a tire that is attached to an overhanging tree limb by a rope. The girl’s father pushes her so that her centripetal acceleration is 3.0 m/s2. If the length of the rope is 2.1 m, what is the girl’s tangential speed?

 b. A boy swings a yo-yo horizontally above his head so

that the yo-yo has a centripetal acceleration of 1.5m/s2. If the yo-yo’s tangential speed is 1.1m/s, what is the length

of the yo-yo?

c. Correct the following statement: The racing car rounds the turn at a constant velocity of 145 km/h.

2.5 m/s

0.81 m

Constant speed not constant velocity

Dynamics of uniform circular motion

Object undergoing uniform circular motion is accelerating with centripetal acceleration ac , so it has a force acting upon it:

2

c c

mv F = ma =

r

which must be directed toward the center of the circle.

It is called centripetal force.

it is not separate force – it is simply one of our familiar forces acting in the role of causing circular motion

Many forces can force an object to move in circular path, therefore becoming centripetal force:

Moon around the Earth ……… gravity

object sitting on a rotating table (strawberries sitting on a seat of a turning car)(a car moving in a circular path) ………… friction

a ball whirling in a circle at the end of a string ……… tension in the string

a person pressed against the inner wall of a rapidly rotating circular room in an amusement park …………. normal force from the wall

Ferris–wheel rider passes through the lowest point of the ride ……. normal force from the seat and the force of gravity

useful relations: - speed(linear) and angular speed

2πRv =

T

2πω =

T

constant speed

period T: time required for one complete revolution (s)

speed v = distance/time

angular speed ω = angle swept/time

Centrifugal (center-fleeing) Force - MISCONCEPTION

Circular Motion -  The radius of a spacecraft orbiting earth is 6.67 x 106 m. If it orbits earth in 5292 seconds, what is the velocity of the spacecraft?7919 m/s  Jimmie Johnson is driving his #48 Lowe’s NASCAR around a bend that has a radius of 70 meters. It takes him 30 seconds to travel the track. What was the centripetal acceleration of Jimmie John’s #48 Lowe’s NASCAR?  3.07 m/s2

 

When a wheel rotates about a fixed axis, do all the points on the wheel have the same tangential speed?Yes Do they all have the same velocity? No

A 1000 kg car is going around a curve with radius 30 meters. If the coefficient of friction between the car's tires and the road is 0.5, what is the maximum speed at which the car can make the turn?

m = 1000 kg

r = 30 m

𝜇=0.5

g = 9.8 m/s2

maximum speed in the turn, v = ?

v = 12 m/s

𝐹 𝑐=𝐹 𝑓𝑟 𝑚 𝑣2

𝑟=𝜇𝐹𝑛