ROTATIONAL MOTION Center of Mass ● Newton's Laws and Conservation Laws – Describe and predict the motion of physical systems – In these laws, a system has only one acceleration ● But systems can be made up of many parts – Each moving and accelerating in different directions – How to decide on one position, velocity, acceleration? ● Center of Mass (“CM”) – The average position of the mass in a system – Newton's Laws describe the motion of the CM – Sometimes, there is no mass at the CM(!) Center of Mass: Example ● Box of air molecules – Many different positions and velocities – CM is at center of box CM – CM of air molecules does not move ● Terminology – Each molecule is said to moving “relative to the CM” ● Momentum and Energy – This system has a total momentum of zero – This system does have Kinetic Energy → Temperature “Rigid” Objects ● The atoms in a rigid object are fixed in position relative to each other – This is an idealization: Every real object will deform if a big enough force is exerted on it ● In physics we often treat solid objects as being “rigid” – We pretend the “atomic springs” are infinitely stiff – A pretty good approximation ● Rigid objects and the CM – Atoms can move relative to the CM only if the object rotates Circular Motion ● Rotation always occurs around an axis – Axis: a line around which the rotation occurs ● When a rigid object rotates around an axis: – Every atom in the object moves in a circle – The radius of an atom's circle is the distance between that atom and the axis Circular Motion ● When a rigid object rotates, every atom takes the same amount of time to complete one circle – This time is called the period of the circular motion – Earth's rotational period is 1 day ● Different atoms move around circles of different radius – So the distance traveled by each atom is different Circumference of circle Speed = ● For any given atom: Period – Atoms which are far from the axis move faster! Angular Speed ● Every circle covers 360° of angle – Every atom in a rotating object covers 360° in one period ● Angular Speed ( symbol: ω ) – Measures how much angle is covered per second – Angular speed is the same for all atoms in a rigid object – ω is also called “rotational speed” ● degrees revolutions radians Units: , , sec sec sec Tangential Speed ● Tangential speed is the actual speed of an atom in a rotating object – It is called “tangential speed” so it doesn't get confused with “angular speed” ● Angular speed is the same for every atom – Tangential speed varies between atoms; it depends on the distance an atom is from the axis of rotation v = tangential speed = v r r = distancefrom axis = angular speed Rotational Inertia ● Concept of “inertia” applies to rotational motion – A rotating object's “natural state” is to continue rotating at a constant angular speed – Applying a force can change rotational speed ● Rotational motion: inertia ≠ mass – Distance from mass to rotation axis is also a factor – Rotational inertia depends on shape – It also depends on the chosen axis Rotational Inertia: An Example ● A spinning figure skater – Begins the spin with arms extended away from the body – Mass of arms is far from axis of rotation – Has a large rotational inertia ● To spin faster, bring arms inward – Decreases rotational inertia – Angular speed increases Torque ● Newton's Laws: Forces can cause changes in speed ● To change angular speed: – Force must be applied (no surprise) – Location and direction of force are important ● The combination of force, location, and direction is called Torque Small Large Torque Torque mg mg Calculating Torque – “Lever Arm” ● Torque has three ingredients: – Strength of the applied force – Direction of the applied force – Location of the applied force ● Direction and location are combined to form a “lever arm” – Distance from the axis of rotation to the location of the force F – Must be measured perpendicular to the direction of the force d Torque = F d Torque Example: See-Saw CW d d CCW 1 2 m g 2 m g 1 ● The torque produced by a person's weight depends on the lever arm ● Torques come in two directions: Clockwise (CW) and Counter-clockwise (CCW) – To balance see-saw, CCW and CW torques must balance perfectly ( zero net torque! ) Torque Example: Accelerating Car CM CM F F Accelerating Car Decelerating Car Forward force makes Backward force makes CCW torque CW torque Front of car tips upward Front of car tips downward Equilibrium: Revised ● A better definition of equilibrium: – F = 0 ( zero net force → constant velocity ) net – T = 0 ( zero net torque → constant angular velocity ) net ● It is possible to have a net force without a net torque! – And vice-versa! Equilibrium and Balance ● A tall object needs a base in order to stand – Edges of the base are made up of the parts of the object in contact with the ground ● Balance of tall objects requires equilibrium – The CCW torque must cancel the CW torque CMCM ● Balance: The CM must be above the base – If the CM moves outside the base: – The object begins to rotate... – And fall down! Using Force to Change Direction ● An object's natural state is a constant velocity – Velocity includes both speed and direction ● A net force changes the velocity of an object – The effect depends on the direction of the net force and the direction of the velocity v v F v F F Speeding up Slowing down Changing direction Centripetal Force ● Circular motion requires changing direction at all times – Therefore it requires a force at all times! ● Centripetal means “toward the center” – The required direction of force for circular motion v v v F F F (road on tires) CCW – constant left turn CW – constant right turn Calculating Centripetal Force ● Centripetal force has three ingredients: – The mass of the object moving along a curve – The tangential speed of the object – The radius of the curve ( tight curves → small radius ) m v2 F = cent r ● Note: Centripetal force doesn't just “happen” – Must be provided by some other force – Tension, friction, gravity, etc. Centrifugal Force (Fictitious) ● Newton's Laws must be used carefully when the observer is accelerating! – If the observer accelerates, “fictitious forces” appear ● Example: The driver of an accelerating car – The driver is pushed forward by the seat ( actual force ) – Observers inside the car feel pushed back into their seat for no reason ( fictitious force – the observer is accelerating ) ● Centrifugal force is a fictitious force felt by observers who are in circular motion ( that is, accelerated ) – Feel an unexplained push toward the outside of the circle Centrifugal Force (Example) ● Soda can on dashboard of car – When car turns left, it is accelerating – Observers in car measure a fictitious force pushing the can to the right Car Can ● A more “Newtonian” view – Can's natural state: move forward in a straight line – When car turns, friction tries to keep can and dashboard together – If friction is too weak to force the can to turn tightly: can goes straight while car goes left Newton's Laws – Rotational Motion ● Analogy: Linear Motion → Rotational Motion – x → θ; v → ω; a → α; m → I; F → T ● Newton's Laws can be applied to rotational motion by replacing the appropriate quantities – 1st Law: Zero net torque → rotation at constant ω – 2nd Law: T = I α – 3rd Law: Action torque → opposite reaction torque Angular Momentum ● The concept of momentum can also be translated into rotational motion: Momentum = m v Angular Momentum = I ● Angular momentum ( like ω ) can point in the CW or CCW direction ● Total angular momentum is conserved Angular Momentum Conservation Example ● As the Moon orbits the Earth: – It pulls on the earth's oceans (causing the tides) – This pull creates friction between the oceans and rocks – This friction slows down the earth's spin ● Earth's angular momentum decreases over time – So the Moon's angular momentum must increase! – Every year, the Moon gets further away by about an inch Summary ● Rigid objects can rotate around their CM ● ω is the same for all atoms, v is different ● Rotational Inertia depends on distance from the axis of rotation ● Rotational motion laws can be made from linear motion laws by simple replacement .