Circular Motion Circular motion occurs about an axis Rotation: object spins about an internal axis Earth rotates about its polar axis once a day. Revolution: object moves about an external axis Earth revolves once about the sun each year.

Speeds involved in Circular Motion Linear speed (v): distance covered per unit time by a point on the object. (m/s) (Also called tangential speed) Rotational speed: amount of angle swept out per unit time. (revolutions per minute (rpm) or radians per second (1/s)) On a rigid rotating object: Is rotational speed everywhere the same? Is linear speed everywhere the same?

For a Rigid Rotating Object Linear velocity is proportional to rotational velocity v ~ r ω r = distance from axis of rotation. “~” means “proportional to” v = rω if ω is in radians per second

Suppose you get a flat tire while driving You put on a “toy spare” tire that came with the car. The toy spare tire is smaller than your other tires. How does this affect your driving?

Period, Frequency and Speed Period (T) : Time for one full rotation or revolution (one round trip). units are seconds (s) Frequency (f) : the number of rotations or revolutions (number of “cycles”) per second. Units are 1/s also called “hertz” (Hz.). Relationship: T=1/f or f = 1/T For constant speed (v), circular motion: Speed = 2π(radius)/period or v = 2πr/T

Some Examples I spin in a swivel chair at a frequency of .5 Hz. What is the period of my spin? A record has a scratch 12 cm from the center that makes the record skip 45 times each minute. What is the linear speed of the scratch?

What causes Circular Motion? Suppose I swing an object at constant speed in a circle. (“uniform circular motion”) Does the object have constant velocity? Does the object accelerate? Does the object feel a force? If so, what causes the force? In what direction is the force? How does the object move if I cut the rope?

Centripetal Force To keep an object in circular motion, we must constantly exert a force Perpendicular to the object’s velocity Directed inward toward the center of the orbit. This direction is called the centripetal direction. The force is called the centripetal force. Examples of centripetal force: Tension in string, keeping ball in orbit. Sun pulling on Earth, keeping it in orbit. Earth pulling on Moon, keeping it in orbit.

Fictitious Forces: Centrifugal Force When a car turns left (inward, “centripetal”), why do you feel pushed to the right (outward, “centrifugal”)? Do you feel like you can stand on the wall of the car? Can we simulate gravity by standing on the wall of a rotating cylinder?

Fictitious Force: Coriolis Effect A force we see due to the rotation of the Earth and how things on Earth move. Foucault Pendulum: proved that Earth Rotates Affects projectile motion Affects flight plans of pilots. Coriolis Effect in Action: Movie From Nasa Wiley Animation and Discussion Wiley discussion2 Effects on Airplane Flight Coriolis Effect on Wind Direction

Centripetal Force (a closer look) There is a relationship between Centripetal Force (FC) and Speed of object (v) Radius of orbit (r) Mass of object (m) FC = mv2/r Let's see a movie of it. Now let’s test it in action.

Centripetal Acceleration Circular motion requires the velocity to constantly change direction. The acceleration causing this change of direction is always directed towards the center. The term “centripetal” means “towards the center”. Centripetal Acceleration: ac= v2/r Centripetal Force: Fc = mac= mv2/r

More Examples Missy’s favorite ride at the fair has a radius of 4.0 meters and takes 2.0 seconds to make one full revolution. What is Missy’s linear speed on the ride? What is Missy’s centripetal acceleration on the ride? The captain of a 60500 kg plane flies in a circle of radius 50.0 km. He completes a circle every 30.0 minutes. What is the centripetal force exerted by the air on the wings of the plane?

Yet More Examples A space station is in a circular orbit about the Earth at an altitude h=500 km. If the station makes one revolution every 95 minutes what are its orbital speed and centripetal acceleration?

2. The Conical Pendulum A mass, suspended from a pivot, swings in a circle as shown. Find its period as a function of r, m, θ, and g.

A Lab We’ll Do. A mass (m1) attached to one end of a string is swung in horizontal circular motion. At the other end of the string, another mass (M2) is suspended. The apparatus is as shown. Find the period of the motion in terms of m1 and M2.

Gravitation Attractive force between two masses (m1,m2) r = distance between their centers of mass.

1. What is the Gravitational Force on an object at the Surface of the Earth? Object has mass (m) Radius of the Earth: RE=6.4*106 m Mass of the Earth: ME=6.0*1024 kg Big G: G = 6.67*10-11 Nm2/kg2

2. An Object in Circular Orbit What is the condition for a stable orbit? Set gravitational force equal to centripetal force.

3. Kepler’s Laws of Planetary Motion (consequence of Newton’s Laws) Planets move in elliptical orbits with the Sun at one focus. A line from the sun to a planet sweeps out equal areas in a given period of time. The square of the orbital period of a planet is proportional to the cube of its average distance from the sun. (T2 ~ r3) (can be derived for circular orbit) Kepler animation *You are not responsible for laws 2 and 3.

4. Planet in Elliptical Orbit about the Sun. Where does a planet have greatest kinetic energy? Where does a planet have greatest potential energy? Where is the planet moving fastest? (HINT: Total Energy is Conserved)

5. Suppose we drill a hole through the Earth and drop someone in. What is the person’s initial acceleration? What is the person’s acceleration at center of the Earth? What happens to his acceleration as he falls towards the center? What happens to his acceleration after he passes the center? Does he reach the other side? If so, then what?

Center of Gravity (CG) Position through which gravity acts on an object if the object were condensed to a particle. CG of an object must be supported to avoid toppling. Stable Equilibrium: Object balanced such that any displacement will raise its CG. (CG will then fall back to lower P.E.) Unstable Equilibrium: Object balanced such that any displacement will lower its CG. (CG will then continue to fall lower P.E.)

Translation, Rotation, and Rolling Translation: motion of the CG only. Rotation: Spinning about the CG. Rolling: A combined motion Object rotates about a constantly moving axis between object and surface. Velocities of points on object are proportional to their distances from the axis.

Rotational Mechanics: Torque Torque causes things to rotate about an axis (just as ________ causes things to ________________). Types of Torque we see everyday: Torsion or twisting: Torque applied about the length of an object. Bending: Torque applied about an axis perpendicular to the object’s length.

What makes up a Torque? Do we need a force? Do we need a net force? Do we need anything else? OR (put another way ) Can I get a torque with no force? Can I get a torque with no net force? Can I apply a force to an object and get no torque?

Requirements for a Torque A Force A Lever Arm, also called a “moment arm”, equals distance from the axis of rotation. The amount of torque (τ) we get depends on the Amount of force we apply (F ┴) Length of lever arm (r) τ = r * F ┴ = Torque about the pivot point

A Balance of Torques? Can we apply a number of torques and have no rotation? Can torques cancel out? A net torque causes rotation. Rotational Equilibrium: τnet= 0 If torque produces counterclockwise rotation it is (+) If torque produces clockwise rotation it is (+)

Examples A meter stick is on a pivot at its center. If a 1 kg mass is placed 8 centimeters to the left of the pivot, what is the torque produced about the pivot? Can I place a .2 kg mass to the right of the pivot and balance the 1 kg mass? If so, where should the .2 kg mass be placed? After placing the .2 kg mass, what is the force exerted by the pivot on the meter stick? What torque does this force produce?

Examples (cont’d) A meter stick is on a pivot at its center. A 1 kg. mass is placed .1 m to the right of the pivot and a .5 kg. mass is placed .2 m to the right of the pivot. Where must a .5 kg. mass be placed to balance these masses? After all masses are in place, what is the force exerted by the pivot on the stick? What is the net torque about the right end of the pivot?

Torque exerted on a spool Which case of applied torque will cause the fastest rolling (for the same applied force)?

Rotational Inertia (I) Resistance of an object to being rotated. It is more difficult to rotate an object about a point if more of its mass is further from that point. It is easier to rotate an object whose mass is closer in to the point of rotation. I ~ mr2 For a small mass, a distance r from a pivot: I = mr2 Ex: Pendulum

Rotational Inertias for various objects. (Note heavy dependence on r)

Angular Momentum Measure of the resistance of an object to having its rotational motion changed. L = I × ω L = angular momentum I = rotational inertia ω = rotational velocity (recall: v = rω) For a mass moving in a circle at speed v: L= I × ω =(mr2) ×(v/r) = r × m v = r × p Applying an external Torque to an object or system: Increases ω and increases L but…..

Conservation of Angular momentum If the net external torque on a system is zero, the angular momentum of the system is constant. Example: L = mvr ; if r decreases with no net torque, then v increases. Figure skaters spin faster when they pull in their arms. Swimmers curl their bodies inward to turn faster after swimming a length. Angular momentum is a vector. If I reverse the direction of spinning, the direction of L reverses.

Torque/Force; Momentum/Angular momentum Here we can see how translational vectors (F and p) relate to rotational vectors (τ and L).