-- Angular momentum. -- Equilibrium. Final Exam. During class (1-3:55 pm) on 6/27, Mon Room: 412 FMH (classroom)

Transcription

1 inal Exam During class (1-3:55 pm) on 6/27, Mon Room: 412 MH (classroom) Bring scientific calculators No smart phone calculators l are allowed. Exam covers everything learned in this course. tomorrow s class 1 hour lecture and 1 hour review session 1 Rotational Motion -- Angular momentum -- Conservation of angular momentum -- Equilibrium 2 1

2 Angular momentum concepts & definition - Linear momentum: p = mv - Angular (Rotational) momentum: L = moment of inertia x angular velocity = Iω linear rotational inertia speed linear momentum m v p=mv I ω L=Iω rigid body angular momentum Angular momentum of a bowling ball 6.1. A bowling ball is rotating as shown about its mass center axis. ind it s angular momentum about that axis, in kg.m 2 /s A) 4 B) ½ C) 7 D) 2 E) ¼ ω = 4 rad/s M = 5 kg r = ½ m I = 2/5 MR 2 L = Iω 2

3 Angular momentum of a point particle 2 L = Iω = mr ω = mvtr = mvr sin θ = mvrp= ptr = prp O r p r vt θ vt v = ω r Note: L = 0 if v is parallel to r (radially in or out) L = I ω = mvt r = mvr sin θ = mvrp = rp = rp sin θ = r p T p Angular momentum of a point particle O r p r p T θ p L= rp = rpsin θ = r p T p If we know components r = ( rx, ry,0) x L = (0,0, rp rp) x y y x p = ( p, p,0) y 3

4 Angular momentum for car 5.2. A car of mass 1000 kg moves with a speed of 50 m/s on a circular track of radius 100 m. What is the magnitude of its angular momentum (in kg m 2 /s) relative to the center of the race track (point P )? A) A B) C) D) E) P B 5.3. What would the angular momentum about point P be if the car leaves the track at A and ends up at point B with the same velocity? A) Same as above B) Different from above C) Not Enough Information L= rp = rpsin θ = r p T p Net angular momentum of particles L = L + L + L

5 Example: calculating angular momentum for particles PP *: Two objects are moving as shown in the figure. What is their total angular momentum about point O? m 2 m 1 τ Δω = Iα = I = Δt ΔL Δt Δ L= τ Δt Conservation of angular momentum for one object : Angular momentum, L is consreved if τ = 0 Conservation of angular momentum for a system of objects Net angular momentum, L is consreved if τ,ext = 0 5

6 A puck on a frictionless air hockey table has a mass of 5.0 g and is attached to a cord passing through a hole in the surface as in the figure. The puck is revolving at a distance 2.0 m from the hole with an angular velocity of 3.0 rad/s. The cord is then pulled from below, shortening the radius to 1.0 m. The new angular velocity (in rad/s) is. Demonstration: Spinning Professor: Web link Isolated System τ, ext = 0 L = constant I ω = I ω i i f f Moment of inertia changes Another link 6

7 Tethered Astronauts 6.3. Two astronauts each having mass M are connected by a mass-less rope of length d. They are isolated in space, orbiting their center of mass at identical speeds v. By pulling on the rope, one of them shortens the distance between them to d/2. What are the new angular momentum L and speed v? A) L =mvd/2 mvd/2, v = v= v/2 B) L = mvd, v = 2v C) L = 2mvd, v = v D) L = 2mv d, v = v/2 E) L = mvd, v = v/2 L = Iω L = mv T r Example: A merry-go-round problem A 40-kg child running at 4.0 m/s jumps tangentially onto a stationary circular merry-go-round platform whose radius is 2.0 m and whose moment of inertia is 20 kg-m 2. ind the angular velocity of the platform after the child has jumped on. 7

8 Equilibrium The Equilibrium Conditions No linear motion and no rotational motion No acceleration, no angular acceleration Net force is zero. Net torque is zero., x = 0, y = 0 τ = 0 (You can choose any axis for the torque calculation.) 8

9 Problem PP : A meter stick balances horizontally on a knife-edge at the 50.0 cm mark. With two 5.0 g coins stacked over the 12.0 cm mark, the stick is found to balance at the 45.5 cm mark. What is the mass of the meter stick? Example 3: A uniform beam, of length L and mass m = 1.8 kg, is at rest with its ends on two scales (see figure). A uniform block, with mass M = 2.7 kg, is at rest on the beam, with its center a distance L / 4 from the beam's left end. What do the scales read? 9

10 Example 2: Beam with a mass, supporting a weight The 2.4 m long 20 kg uniform beam shown in the figure is supported on the right by a cable that makes an angle of 50 o with the horizontal beam. A 32 kg mass hangs from the beam 1.5 m from the pivot point on the left. x m = 32 kg Determine the cable tension needed to produce equilibrium θ = 50 o, x = 1.5 m, L = 2.4 m ind the forces on the pivot 8.1. The sketches show four overhead views of uniform disks that can slide or rotate on a frictionless floor. Three forces act on each disk, either at the rim or at the center. Which disks are in equilibrium? A) 1, 2, 3, 4 B) 1, 3, 4 C) 3 D) 1, 4 E) 3, 4 = 0 = 0 τ 10

11 A uniform block with 2 m x 3 m rectangular cross section is on a rough inclined surface, as shown below. The surface is very rough so that the block does not slide, but the block can tip over if theta is too large. ind the maximum angle theta before it tips over. 3 m 2 m θ 11