Physics of Ramps/Inclined Planes: How to move your piano up without breaking your back W Floor Why doesn’t the piano sink into the floor? Floor exerts a support/contact force N on the piano, equal to weight W. N

If you push on any object. It will push back on you with the same amount of force. (floor pushes back up on piano legs) Newton’s 3 rd Law of Motion (Action- Reaction): For every force that one object exerts on a second object, there is an equal but oppositely-directed force exerted by the second object on the first. Examples: An ‘explosion’ Rocket Propulsion Force exerted by skater on bus = Force exerted by bus on skater. Skater moves faster because of her smaller mass. Force exerted by rocket on expelled gases = Force exerted by expelled gases on rocket

So far, we have been adding forces in 1 dimension. What if they point in different directions ? Being vectors, forces add geometrically.

Going back to the Piano Problem…. Solution 1: We can lift the piano to the required floor height h piano mg = 2000 N F piano mg h = 2 m F We clearly need to do physical Work to move piano up W = (Force)( distance ) = mgh Where does this work go ? It is stored in the ‘Potential Energy’ U of the piano. U = mgh (gravitational potential energy) F = mg (a) (b) Situation (b) Has to have more ‘energy’ stored in it than (a). Energy = capacity to do physical work (think of it as a currency that remains constant in any physical ‘transaction’; can’t be created/destroyed) Note: Energy could be potential or kinetic (contained in motion)

Solution 2: We can slide the piano along a ramp. mg N 200 N h = 2 m The ramp supplies most of the force needed to keep it from ‘falling’. You supply the smaller amount needed to keep it from sliding. Compare the amount of work you do in both solutions. - You supply less force in (b) but need to push through a distance greater than h. Work = larger force small distancein (a) = smaller force larger distance in (b) Ramp has mechanical advantage

So far, we have been treating matter as being pointlike balls and ignoring the fact that they are extended bodies, behaving in more complicated ways Point-like bodies can only Translate. Real extended bodies can also Rotate (about some axis). Motion of Flying balls and point-like objects ? Get real!

W W’ What makes a seasaw rotate ? What observations can we make about its rotation ? Claim: There must be an analogy between translation and rotation Flashback to childhood : The Seasaw

Motion of An ‘Isolated’ Seasaw If initially stationary ……must remain stationary. If initially rotating at some rpm…..must continue to turn at the rpm WHY ?Because of Inertia……or ROTATIONAL INERTIA What gives a measure of inertia ? Mass A massive wheel is harder to turn than a light wheel. Distribution of MassThe closer mass is to the axis of rotation, the easier it is to turn. The further out, the harder it is to turn

The Rotating Isolated Seasaw … Like the isolated skater who is coasting….the isolated seasaw turns constantly  O oo  To describe rotation, one needs to define : 1. the axis of rotation and 2. the angular quantity, using the Right Hand Rule) We will use axis O (pointing out of the page)  = angular position (in radians; 2  radians in 360  )  =  /  t (rad/s), the angular velocity, about O Question: What will happen if you add more spin (by some external influence called Torque) ?

Torque will cause the seasaw’s rotational velocity to change !  O oo  a=  /  t (rad/s 2 ), angular acceleration about O Rotational Analog of Newton’s Second Law:T = I  Torque Moment of Inertia Angular Acceleration What does it mean? Spinning a marble is easier than spinning a bowling ball. To change the spin of a top, you need to apply torque. How does one apply torque ? One applies a Force F with a lever arm r.

Torque = (lever arm)( Force) or T = r x F 2 m 1 m 200 N 400 NO m Axis of rotation What happens if m is placed beside Red Box ?

Analogy between Translation and Rotation TranslationRotation Linear Displacement  xAngular displacement  Linear Velocity vAngular speed  =  /  t Linear Acceleration a =  v/  tAngular acceleration  =  /  t F = maT = I  Mass mMoment of Inertia I (depends on mass, mass distribution) Translational Kinetic Energy Rotational Kinetic Energy = (1/2) mv 2 = (1/2) I  2

Everyday Examples of Rotational Motion

Special Case to Analyze: Uniform Circular Motion (constant  ) Whirling a String of Light Bulbs at constant angular speed  v1v1 v2v2 v3v3 v4v4 Q: What keeps bulbs in circular orbit ? A: Tension of String (a central force), which points to the center. What happens if you cut the string ? How will the bulbs fly?

How large is the central (centripetal) acceleration ?