Uniform Circular Motion
Variables of Circular Motion
Velocity and Acceleration in Uniform Circular Motion
In Uniform Circular Motion happens when an object moves in a circle with constant speed but changes direction.
Demonstration
Question 1
How do we know the stopper is accelerating?
Question 2
What is the direction of the acceleration of the stopper?
Question 3
What force is producing the centripetal acceleration of the stopper?
Question 4
What is the direction of the net force on the stopper?
Example 1
The bobsled track at the 1994 Olympics in Lillehammer, Norway, contained turns with radii of 33 m and 24 m, as in the figure. Find the centripetal acceleration at each turn for a speed of 34 ms, a speed that was achieved in the two-man event. Express the answers as multiples of g.
Example 2
The model airplane in the figure has a mass of 0.90 kg and moves at a constant speed on a circle that is parallel to the ground. The path of the airplane and its guideline lie in the same horizontal plane, because the weight of the plane is balanced by the lift generated by its wings. Find the tension T in the guideline (length = 17 m) for speeds of 19 m/s and 38 m/s.
Example 3
A 1200
kg car rounds a corner of radius r = 45 m. If the coefficient of static
friction between the tires and the road is 
,
what is the greatest speed the car can have in the corner without skidding?
Question 5
When the ball reaches the break in the circle, which path will it follow?
Angular Displacement and Angular Velocity
Angular Displacement – Angular displacement of a body is the angle in radians through which a point or line has been rotated in a specified sense or direction.
Angular Velocity – Angular velocity is angular displacement divided by elapsed time during which the displacement occurs.
Angular speed – 
Speed, Velocity, and Acceleration in Uniform Circular Motion
Velocity and angular speed
Example 4
The disk in a hard drive in a desktop computer rotates at 7200 rpm. The disk has a diameter of 5.1 inches (13 cm). What is the angular speed of the disk?
Acceleration
Example 5
The hard drive disk in the previous example rotates at 7200 rpm. The disk has a diameter of 5.1 inches (13 cm). What is the speed of a point 6.0 cm from the center axle? What is the acceleration of this point on the disk?
Dynamics of Uniform Circular Motion
Example 6
In the track and field event known as the hammer throw, an athlete spins a heavy mass in a circle at the end of a chain. Once the mass gets moving at a good clip, the athlete lets go of the chain. The mass flies off in a parabolic arc; the winner is the one who gets the maximum distance. For male athletes, the hammer is a mass of 7.3 kg at the end of a 1.2 m chain. A world-class thrower can get the hammer up to a speed of 29 m/s. If an athlete swings the mass in a horizontal circle centered on the handle he uses to hold the chain, what is the tension in the chain?
Example 7
A level curve on a country road has a radius of 150 m. What is the maximum speed at which this curve can be safely negotiated on a rainy day when the coefficient of friction between the tires on a car and the road is 0.40?
Example 8
If a roadway is banked at the proper angle, a car can round a corner without any assistance from friction between the tires and the road. Find the appropriate angle for a 900 kg car traveling at 20.5 m/s in a turn of radius 85.0 m.
Example 9
While driving along a country lane with a constant speed of 17.0 m/s, you encounter a dip in the road. The dip can be approximated as a circular arc, with radius of 65.0 m. What is the normal force exerted by a car seat on a 80.0 kg passenger when the car is at the bottom of the dip?
Example 10
A car of mass 1500 kg goes over a hill at
a speed of 20 m/s. The shape of the hill is approximately circular, with
a radius of 60 m, as in the figure. When the car is at the highest point
of the hill,
(a) What is the force of gravity on the car?
(b) What is the normal force of the road on the car at this point?
Example 11
The centrifuge in the figure rotates at a rate that gives the bottom of the test tube a linear speed of 89.3 m/s. If the bottom of the test tube is 8.50 cm from the axis of rotation, what is the centripetal acceleration experienced there?
Example 12
In a 1901 circus performance, Allo "Dare Devil" Diavolo introduced the stunt of riding a bicycle in a loop-the-loop. Assuming that the loop is a circle with radius R = 2.7 m, what is the least speed v Diavolo could have at the top of the loop to remain in contact with it there?
Circular Orbits and Weightlessness
For a projectile close to the Earth
Now using centripetal acceleration
So the orbit velocity
Period of a satellite
Example 13
Phobos is one of two small moons that orbit
Mars. Phobos is a very small moon, and has correspondingly small gravity
– it varies, but a typical value is about 
.
Phobos isn't quite round, but it has an average radius of about 11 km.
What would be the orbital speed around Phobos, assuming it was round with
gravity and radius as noted?
Newton's Law of Gravity
Example 14
A typical bowling ball is spherical, weighs 16 pounds and has a diameter of 8.5 in. Suppose two bowling balls are right next to each other in the rack. What is the gravitational force between the two – magnitude and direction?
Example 15
As part of a
daring rescue attempt, the Millennium Falcon passes between a pair of
twin asteroids, as shown. If the mass of the spaceship is 
and
the mass of each asteroid is 
,
find the net gravitational force exerted on the Millennium Falcon (a)
when it is at location A and (b) when it is at location B. Treat the
spaceship and the asteroids as if they were point objects.
Gravity on Other Worlds
Gravity and Orbits
Example 16
A spacecraft is orbiting the moon in an orbit very close to the surface – possibly because of the moon's lack of atmosphere. What is the craft's speed? The period of its orbit?
Kepler's Law
Example 17
Phobos is the closer of Mar's two small
moons, orbiting at 9400 km from the center of Mars, a planet of mass 
.
What is Phobos's orbital period? How does this compare to the length of
the Martian day, which is just shy of 25 hours?