Ch+7+-+Rotation

Chapter 7 Rotational Motion

Ø (theta) = Angular position which was previously represented by x w (omega) = Angular Velocity which was previously represented by v a (alpha) = angular acceleration which was previously represented by a

Angular definitions

Ø = L/r Theta is equal to the arc length divided by the radius

w = delta Ø/ t Omega is equal to the change in theta divided by the time

a = delta omega/t Alpha is equal to the change in omega divided by the time

To convert from radians to degrees you multiply 1 radian * (360 degrees/2Pi)

Conversion factor 1 revolution is equal to two pi radians ex) (360 rev/min) * (1 min/60s) * (2Pi rad/1 rev)

Angular Equations w (final) = w (initial) + at Ø = w (initial)*t +( ½*a* (t^2)) w^2 = (w (initial) ^2) + 2aØ

Linear - Angular relationship ( V tangent) = rw (a tangent) = ra

The merry go round effect The further out you are the faster you go because your speeding up going around the circle. The inner circle is traveling slower.

Objects travel in circular path at constant speed v = (2*Pi*r)/T T = the period which is the time is takes to complete 1 revolution. The periods unit is seconds. Objects want to travel in a straight line (Newton's 1st law)

Circular motion - constant speed

Objects do not travel in circular paths. they must be forced into circular paths. centripetal acceleration = (v^2)/(r) = (4*(Pi^2)*r)/(T^2) centripetal acceleration =r(w^2) Have acceleration only if going in a circle

Direction of vectors Velocity - Tangential Acceleration - Radially inward

The Right Hand Rule - Clockwise means into the page. Think of the clock going into the wall. This tells you the direction of the vector.

**MERRY-GO-ROUND APPLICATION** On a merry-go-round, if you were to stand closer to the center of the merry-go-round, the linear velocity would be less than if you were to stand closer to the outside of it. This is because the person closest to the outside of the ride would have to travel further, due to the larger circumference, in order to make one full revolution. However, the angular speed is constant anywhere on the merry-go-round because it does not involve distances(Refer to figure 1 below).    

Typically, the younger the kids that ride the merry-go-round, the closer to the center they sit. This is because of the centripetal force being exerted on them. This force is created by the rotation of the platform of the ride. Centripetal force is the force pushing outward, away from the center. The further from the center one is, the greater the centripetal force and is and likelihood of them flying off the ride. But on the inside, the person can usually let go and not go anywhere because the centripetal force is not great enough. The contradicting force of centripetal force is friction. Friction allows one to stay place despite the centripetal force. It is the only factor keeping the person on the merry-go- round. Adults and older kids tend to be on the outside of the merry-go-round because they can a greater source of friction, since they are stronger, whereas the kids on the inside of the ride don't have to be as strong and can ride more comfortably.

**Gravitation**

Law of Gravitation - every object exerts a force on every other object -this force is directly proportional to product of masses and inversely proportional to the square of distance of separation In formula speak: and

Refer to the following illustration for the next formula:

When solving for the force of gravitation between two objects, the constant G is multiplied by both masses. The product is then divided by the quare of the distance separating the two object.

Formula for Gravitation:

Value of G:

Relating the Quantities Because Fg is directly proportional to the product of the masses, doubling the mass of one of the objects will double the force of gravitation between the two objects. Similarly, because Fg is inversely proportional to the square of the distance between the two objects, doubling the distance between the two objects will quarter the force of gravitation between the two objects. (this is because distance is squared).

Explained Visually:

Ptolemy initially theorized that the earth was the center of the universe. This was accepted for centuries, though Kepler challenged the theory. Later, Kepler would observe planetary motion to prove it, even though some of his data has been proven false.
 * Planetary Motion**

Kepler's Laws: 1) Planets travel in an elliptical path around the sun. 2) A line from the sun to the planet will sweep out equal area for equal time. 3) ; R=average orbital radius T=period (time for one revolution)

Planetary Data: -Earth is 1 AU (astronomical unit) away from the sun -Earth's period is one year -Moon is 3.8x10^8 meters from the earth -Moon's period is 27.3 days -constant's are relative; the objects must be going around the same things

Planetary Data: Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune D. From Sun (km) 5.79E7 1.08E8 1.50E8 2.28E8 7.78E8 1.43E9 2.87E9 4.50E9 Mass (kg) 3.30E23 4.87E24 5.98E24 6.42E23 1.90E27 5.69E26 8.70E25 1.03E26

Gravity forces objects into a curved path.

Newton's Proof of the Law of Gravitation: This is the gravitation for the sun only!

Weighing the Earth: The following equations were used to solve for the mass of the earth; the final equation can be used in any relevant situation to solve for any of the including variables.

Speed of Object in Orbit

Period of an Orbiting Object

a = F/m or F = m*a || Newton's 2nd: rot. a = torque / I, or torque = I*(rot. a) || m * v || rotational momentum = I * (rot. v) || (1/2) m v2 || rotational kinetic energy = (1/2) I (rot. v) ||
 * ====__For translational motion:__==== || ====__For rotational motion:__==== ||
 * position, distance, x || angle ||
 * velocity, v || rotational velocity ||
 * acceleration, a || rotational acceleration ||
 * inertia (measured by mass), m || rotational inertia (measured by mass and the distribution of mass), I ||
 * force, F || torque (rotational force * the lever arm length) ||
 * Newton's 2nd:
 * momentum =
 * kinetic energy =