Exam Results. Your scores will be posted before midnight tonight. Score Range = Approx. le<er Grade = A = B = C = D 0 34 = F

Transcription

1 Exam Results Your scores will be posted before midnight tonight. Score Range = Approx. le<er Grade = A = B = C = D 0 34 = F Mean 60% Std 20% (You can pickup your exam a1er class on Wednesday or at office hours tomorrow)

2 Today Finish Ch 5 and Start Ch 6 Today Finish Chapter 5 o Circular mo%on o Dynamics of circular mo%on Chapter 6 o Work done by forces o Kine%c energy o Work and kine%c energy with non- constant forces

3 Example: Fric%on w/ accelera%on Block A (mass 2.25kg) rests on a tabletop. It is connected by a light cord passing over a fric%onless pulley to a hanging block B (mass 1.30kg). The coefficient of kine%c fric%on between black A and the tabletop is μ =.450. ATer the blocks are released from rest find the speed of each ater 3.00 cm and the tension in the cord. f s µ s n, f k = µ k n

4 Example: Fric%on w/ accelera%on cont. f s µ s n, f k = µ k n

5 Dynamics of circular mooon v r y R θ r Radial or Centripetal accelera%on a r = r 2 ˆr = v2 r ˆr r x The period is the %me it takes to make one full cycle For constant speed then T = circumference speed = 2 R v v = 2 R T uniform circular mo%on only a rad = v2 R = 4 2 R T 2 F net = ma rad = m v2 R

6 Dynamics of circular mooon If a par%cle is in uniform circular mo%on, both its accelera%on and the net force on it are directed toward the center of the circle. The net force on the par%cle is F net = mv 2 /R. If the string breaks, no net force acts on the ball, so it obeys Newton s first law and moves in a straight line.

7 Non- uniform circular mooon For non- uniform circular mo%on the speed is varying, so we no longer have a constant speed around the circular path. The accelera%on vector can have a tangen%al component ( radial component ( ) Direc%onal change of gives Magnitude change of gives The total accelera%on is a rad = a v v a " = # v2 r ˆr a = d v dt ˆ a = a + a a = a 2 + a 2 a tan = a ) and a a tot a tan a rad

8 Example Conical pendulum An inventor designs a pendulum clock using a bob with mass m at the end of a wire of length L. The bob is to move in a horizontal circle with constant speed v, with the wire making a fixed angle β with the ver%cal direc%on. This is called a conical pendulum because the wire traces out a cone. Find the tension F in the wire and the period T. Remember 2π R v = T 2 2 v 4π R arad = = 2 R T mv Fnet = marad = R 2

9 Example - Conical pendulum cont. Find the tension F in the wire and the period T. Remember 2π R v = T 2 2 v 4π R arad = = 2 R T mv Fnet = marad = R 2

10 Clicker QuesOon A sled moves on essen%ally fric%onless ice. It is a`ached by a rope to a ver%cal post set in the ice. Once given a push, the sled moves around the post at constant speed in a circle of radius R. If the rope breaks A. the sled will keep moving in a circle. B. the sled will move on a curved path, but not a circle. C. the sled will follow a curved path for a while, then move in a straight line. D. the sled will move in a straight line.

11 Example Mo%on in a ver%cal circle A small car with mass M travels at a constant speed, v, on the inside of a ver%cal track with radius R. If the normal force exerted by the track on the car when it is at the top is N A, what is the normal force, N B, on the car when it is at the bo`om of the track? v R v

12 Example Rounding an banked curve For a car traveling at a certain speed, it is possible to bank a curve at just the right angle so that no fric%on at all is needed to maintain the car s turning radius. Then a car can safely round the curve even on wet ice. You need to rebuild a curve so that a car moving at a chosen speed v can safely make the turn even with no fric%on. At what angle β should the curve be banked?

13 Clicker QuesOon A pendulum of length L with a bob of mass m swings back and forth. At the low point of its mo%on (point Q), the tension in the string is (3/2)mg. What is the speed of the bob at this point? Remember v = 2 R T a rad = v2 R = 4 2 R T F net = ma rad = mv2 R a) b) c) d) e) 2 gl 2gL gl gl 2 gl 2 P Q R

14 Energy, Work and Power Principle of conserva%on of energy Energy cannot be created nor destroyed - > E=mc 2 Chapter 6 Types of Energy Kine%c Poten%al Spring Chemical Thermal/solar/nuclear Before I fall I have poten%al energy but no kine%c energy I have lots of kine%c energy

15 Work Physics defini%on: W r 2 r 1 F " d r # Units of Joules: Nm=J Work is a scalar value that is determined by two vectors: remember the dot product B Ɵ A B Ɵ A A B = A(Bcos) C = A B = B A = ABcos B A = (Acos)B A B = A x B x + A y B y + A z B z

16 Work with constant force If the force is a constant, then W = " F d r = F " = F ( r 2 # r 1 ) = F $ r W = F s = F s = (F cos)s Do not memorize Fscos θ d r θ F F = F cos θ S Parallel components of the force to the mo%on mul%ply. If there is no mo%on then the work being done is zero.

17 FricOon and work Fric%on does nega%ve work since it always opposes the direc%on of displacement Its doing nega%ve work which means it is taking energy away from an object For example: You pull on a box with a force of 25N and the fric%on of the box with the floor is 8N. What is the work done by all forces ater the block moves 30cm? The total work done is the sum of all the individual quan%%es of work. f k =8N T = 25N s = 30cm

18 Work- energy theorem The total work done is related to the change in a body s posi%on but it is also related to the change in the speed of a body. F a = F / m = const. Start with constant then We can use the equa%ons of mo%on for constant accelera%on: ( ) v 2 f = v 2 i + 2 a x f x i " v 2 f v 2 F % i = 2$ ' s # m & 1 2 m v 2 2 f v i ( ) = F ( s = W And if we define the kine%c energy as K = 1 2 mv2 Then the work equa%on becomes Work- energy theorem W = 1 2 m(v 2 f v 2 i ) = K f K i = "K

19 Clicker QuesOon Two iceboats (one of mass m, one of mass 2m) hold a race on a fric%onless, horizontal, frozen lake. Both iceboats start at rest, and the wind exerts the same constant force on both iceboats. Which iceboat crosses the finish line with more kine%c energy (KE)? A. The iceboat of mass m: it has twice as much KE as the other. B. The iceboat of mass m: it has 4 %mes as much KE as the other. C. The iceboat of mass 2m: it has twice as much KE as the other. D. The iceboat of mass 2m: it has 4 %mes as much KE as the other. E. They both cross the finish line with the same kine%c energy.

20 Back to simple example f k =8N 3kg T = 25N s = 30cm Find the velocity of the box at 30cm we have two op%ons to solve this Old Way: F x = ma x v f 2 = v i 2 + 2a x "x Or New Way: W = K F net " s = 1 2 mv 2 f # 1 2 mv i2

21 Prelecture quesoon 1 and Clicker QuesOon An apple is ini%ally sisng on the bo`om shelf of a pantry. A hungry physics student picks up the apple to eat it, but changes her mind and puts the apple down on a shelf somewhere above its original loca%on. During this process, the total work done on the apple by all forces is: a) Posi%ve b) Nega%ve c) Zero

22 Prelecture quesoon 2 A box is ini%ally sliding across a fric%onless floor toward a spring which is a`ached to a wall. The box hits the end of the spring and compresses it, eventually coming to rest for an instant before bouncing back the way it came. The work done by the spring on the box as the spring compresses is: a) Posi%ve b) Nega%ve c) Zero

23 Checkpoint quesoon 1 Three objects having the same mass begin at the same height, and all move down the same ver%cal distance H. One falls straight down, one slides down a fric%onless inclined plane, and one swings on the end of of a string. In which case does the object have the biggest total work done on it by all forces during its mo%on? a) Free Fall b) Incline c) String d) Same

24 Checkpoint quesoon 2 A box sits on the horizontal bed of a truck accelera%ng to the let as shown. Sta%c fric%on between the box and the truck keeps the box from sliding around as the truck drives. The work done on the box by the sta%c fric%onal force as the accelera%ng truck moves a distance D to the let is? a) Posi%ve b) Zero c) Nega%ve d) Depends on the speed of the truck.

25 Checkpoint quesoon 3 A car drives up a hill with constant speed. Which statement best describes the total work done on the car by all forces as it moves up the hill with a constant speed? a) Posi%ve b) Zero c) Nega%ve

26 Varying Forces Up %ll now we have been assuming force is constant. Graphically we can interpret W=Fs F If instead we have a non- constant force and we look at the F vs x curve. Graphically the work done is s%ll the area under the curve. W # r 2 r 1 F " d r This will be true for any force and any curve. F S S

27 Clicker QuesOon Three objects having the same mass begin at the same height, and all move down the same ver%cal distance H. One falls straight down, one slides down a fric%onless inclined plane, and one swings on the end of a string. What is the rela%onship between their veloci%es when they reach the bo`om? H Free Fall Fric%onless incline String A) v f > v i > v p B) v f > v p > v i C) v f = v p = v i