1) Law of Orbits - all planets move in elliptical orbits with the Sun at one focus

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2 1) Law of Orbits - all planets move in elliptical orbits with the Sun at one focus 2) Law of equal areas - ANGULAR MOMENTUM IS CONSERVED A line that connects a planet to the Sun sweeps out equal areas in equal time. 3) The Law of Periods : The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit 3 r sat = GM T 2 4π 2 T 2 = 4π 2 3 r sat GM

3 p at h = p 0 + ρgh Pascal s Principle Pressure applied to a confined fluid increases the pressure throughout by same amount p out = p in F out = F in A out A in Demo

4 Buoyancy and Archimedes Principle Buoyant Force - is equal to the weight of fluid displaced by the object - is directed UPWARDs F B = m F g UPWARD - Does not depend on the shape of the object ONLY volume. - Applies to partially or completely immersed object A stone drops F B mg ( ) < 0 a downward A bag of water stays put ( F B mg) = 0 a = 0 Wood will rise ( F B mg) > 0 a upward

5 Buoyancy and Archimedes Principle Buoyant Force - equal to the weight of fluid displaced by the object - is directed UPWARDs - Does not depend on shape of object - ONLY volume. - Applies to partially or completely submerged object F B = m F g UPWARD Which has larger buoyant force? Sink or float? A 200-ton ship is in a tight-fitting lock so that the mass of fluid left in the lock is much less than the mass of the ship. Does it float? 1. No. The ship touches the bottom since it weighs more than the water. 2. Yes, as long as the water gets up to the ship s waterline. If the volumes are the same, they displace the same mass of fluid so the buoyant forces are the same The answer is 2. What matters is the mass of the displaced fluid.

6 Clicker Question You are on a distant planet where the accelera2on due to gravity is half that on Earth. Would you float more easily in water on that planet? 1. Yes, you will float higher 2. Floa2ng would not be changed. 3. No, you will float deeper. FBD of system: F B W Weight (downward) = m object g Buoyant force (upward) = m displaced g

7 Archimedes Principle : Apparent weight in a fluid Weigh in air, then weigh in water. From this, and knowing ρ water you can get object s ρ. F B ma = Weight apparent = Weight actual F mg B Because of buoyant force, apparent weight in water is less than actual weight Archemedes : Is the King s crown gold??? (Hiero III BC) Is it gold??? W actual W actual W apparent = ρ crownvg ρ water Vg = ρ crown ρ water Specific gravity Weight apparent = Weight actual F B = ρ crown Vg ρ water Vg 14.7 kg =11.3??? 14.7 kg 13.4 kg Specific gravities: Gold (Au) 19.3 Lead (Pb) =14.7 F B ρ water Vg =1.3N V = m 3 ρ crown = 14.7 Vg =11300 kg m 3

8 Specific Gravity specific gravity = ρ ρ WATER Rather than using the large SI unit it is sometimes convenient to use specific gravity Material air water Al Fe Cu Pb Os ρ (kg/m 3 ) Specific gravity

9 Helium Blimps Length 192 feet Width 50 feet Height 59.5 feet Volume 202,700 cubic feet (5740 m 3 ) Maximum Speed 50 mph Cruise Speed 30 mph Powerplant: Two 210 hp fuel-injected, air-cooled piston engines ρ He = kg/m 3 & ρ air = 1.21 kg/m 3 What is the maximum load weight of blimp (W L ) in order to fly? F B At static equilibrium F = 0 : W He + W L = F B W L W He The fluid blimp is in is: air W L = m air g m He g W L = F B W He = ρ air V ship g ρ He V ship g = V ship g( ρ air ρ He ) = (5740 m 3 )(9.8 m /s 2 )( kg/m 3 ) = 58 kn = 13,000 lbs Maximum Gross Weight 12,840 pounds

10 Previous Exam Problem What we know: F top = F bottom p top A top = p bottom A bottom p ( 2 top πr ) 2 = ( p top + ρ water gh) ( 2 πr ) 1 p top ( π( R 1 + h cot60) 2 ) = ( p top + ρ water gh )( 2 πr ) 1

11 A cubical block of something, 10.0 cm on a side, floats at the interface between oil and water with its lower surface 1.50 cm below the interface (see the cross section view in the figure). The density of the oil is 790 kg/m 3. (a) What is the gauge pressure at the upper face of the block? (b) What is the gauge pressure of the lower face of the block? (c) What is the mass and density of the block? a) b) Previous Exam Problem Let the top be at a distance y 0 from the top surface Let the length of a side be L Let h be distance from the top of the block and the interface Let h oil be the position of the interface. Δp top surface = ρ oil gy 0 Δp lower surface = ρ oil gh oil + ρ water g(l hy 0 ) c) m block g = F buoyant = F B (oil) + F B (water) ρ block L 3 g = ρ oil L 2 y 0 g + ρ water L 2 (L y 0 )g ρ block = y 0 ( ) ρ oil ρ water L + ρ water ρ block = 1.5 ( ) = 968.5km / m 3 10

12 Chapter 15 : Oscillations Mo2ons that Repeat themselves in a well prescribed way (i.e. period) Swing a ball back and forth with certain frequency f Moving a mass m back and forth on a fric2onless with certain angular frequency ω x KinemaIcs: posi2on: x(t) velocity: v(t) accelera2on: a(t) Dynamics: driving force energy

13 OscillaIons characterisics Frequency f (# of oscilla2ons/second) Units: hertz = 1 Hz = 1 oscilla2on/s = 1 s 1 Period T (2me for one oscilla2on cycle) = 1/f 2x m 1 period = 1 cycle Don t forget to put your calculator in RADIAN mode

14 Simple Harmonic MoIon (SHM) KinemaIcs PosiIon Velocity AcceleraIon In SHM, the accelera2on is propor2onal to the displacement but opposite sign. The propor2onality is the square of the angular frequency.

15 RelaIonships for SHM How to determine x max and φ : INITIAL CONDITIONS Knowing x(0) and v(0),

16 SHM and Hooke s Force Law (again ) Dynamics In SHM, the accelera2on is propor2onal to the displacement but opposite sign. The propor2onality is the square of the angular frequency. Block spring is linear SHO: NOTE: Independent of amplitude Larger mass > longer period

17 Question 15-1 A mass (m) is ahached to one end of a spring (k) and is free to slide on a fric2onless horizontal surface. The mass is pulled back from its equilibrium posi2on and released from rest. What is the accelera2on of the mass at the equilibrium posi2on (x = 0)? 1. At its maximum value 2. It depends on the spring constant 3. It depends on the mass 4. Zero

18 Example A block whose mass m is 680 g is fastened to a spring whose constant k is 65 N/m. The block is pulled a distance x = 11 cm from its equilibrium posiion at x = 0 on a fricionless surface and released from rest at t = 0. (a) What are the angular frequency, frequency, and period of the resuling moion? (b) What is the phase angle and amplitude of the oscillaion? (c) What is the maximum speed and acceleraion?