Chapter 7: Energy. Consider dropping a ball. Why does the ball s speed increase as it falls?

Transcription

1 Chapter 7: Energy Consider dropping a ball. Why does the ball s speed increase as it falls? Viewpoint #1: Force of gravity causes acceleration which causes velocity to change. Viewpoint #2: Force of gravity does work on ball, giving it kinetic energy. New Viewpoint: Force convert s the ball s gravitational potential energy into kinetic energy as it falls down. Ball contains a different form of energy, potential energy, that depends on the particle s position. This concept emerges from conservative forces (forces in which the work done only depends on the initial and final positions)

2 Gravitational Potential Energy Recall that the work done by gravity is proportional to the change in the object s height: Consider an object of mass m going from point A to point B. The work done by gravity is W grav = A B F dl = y 2 y 1 ( mg) dy = mg(y 1 y 2 ) Notice this work is independent of the trajectory. It only depends on the final and initial positions. (Gravity is a conservative force)

3 Gravitational Potential Energy Let s define the change in gravitational potential energy as negative work done by gravity: W grav = Thus the change in gravitational potential energy is U grav = U(B) U(A) =mgy 2 mgy 1 If gravity is the only force that does work, then the work-kinetic energy theorem gives W grav = U grav = K Rearranging: U grav (K + U grav )= E =0 ) K(B)+U grav (B) =K(A)+U grav (A)

4 Q7.1 Clicker Question A piece of fruit falls straight down. As it falls, A. the gravitational force does positive work on it and the gravitational potential energy increases. B. the gravitational force does positive work on it and the gravitational potential energy decreases. C. the gravitational force does negative work on it and the gravitational potential energy increases. D. the gravitational force does negative work on it and the gravitational potential energy decreases.

5 Zero-Point for Potential Energy U grav = U(B) U(A) =mgy 2 mgy 1 Potential energy is a function of position. U(y 2 ) U(y 1 )=mg(y 2 y 1 ) Note: There is no equation that gives the absolute potential energy-just the difference. This means we can define any zero point we choose. Let us choose U(y 1 =0) = 0. Thus U grav (y) =mgy Of course you are free to pick any location for the choice of y=0! Pick one that is convenient for the problem and stick with it consistently.

6 Consider a rock thrown up from the ground. The sum of the potential energy plus kinetic energy is the same at all locations.

7 Clicker Question You toss a kg baseball straight upward so that it leaves your hand moving at 20.0 m/s. The ball reaches a maximum height y 2. What is the speed of the ball when it is at a height of y 2 /2? Ignore air resistance. A m/s B. less than 10.0 m/s but more than zero C. more than 10.0 m/s D. not enough informa<on given to decide v 2 = 0 v 1 = 20.0 m/s y 2 y 1 = 0

8 Chalkboard Question Christine runs forward with her sled at 2.0 m/s. She hops onto the sled at the top of a 5.0 m high, very slippery slope. What is her speed at the bottom? E o = E f 1 2 mv2 0 + mgy 0 = 1 2 mv2 f q v f = v0 2 +2gy 0 = 10 m/s

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11 Clicker Question A cart rolls without fric<on along a track. The graph of PE vs. posi<on is shown. The total mechanical energy (KE + PE) is 45kJ E_tot PE(kJ) x(m) To within 5 kj, what is the maximum KE over the stretch of track shown? A) 25 kj B) 10 kj C) 45 kj D) 35 kj E) None of these

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13 Chalkboard Question (YF 7.32) A car in an amusement park ride rolls without friction around the track. It starts from rest at point A at a height h above the bottom of the loop. What is the minimum value of h such that the car moves around the loop without falling off at point B?

14 Chalkboard Question A pendulum consists of a string attached to a ceiling and a 10 kg bowling ball attached to the other end. The ball is released from rest while touching the ceiling. If the maximum tension of the string is 200 N, at what angle between the string and the vertical direction will the string break?

15 Challenge Problem: A ring of mass M hangs from a thread, and two beads of mass m slide on it without friction. The beads are released simultaneously from the top of the ring and slide down opposite sides. Find the angle between each bead and the vertical such that the ring begins to accelerate upward.

16 Conservative Forces and Conservation of Energy So far we have only considered situations involving gravitational potential energy. We can attribute a potential energy for ANY conservative force. W = A conservative force is a force in which the work done is independent of the trajectory in going from point A to point B. Thus the work depends only on the endpoints. Z B A ~F ~dl U = U(B) U(A) = W

17 Conservation of Energy for Multiple Conservative Forces Suppose there are several conservative forces acting on an object. Again, using the work-kinetic energy theorem W net = K i E = K + i W i = K ( U i )= K i U i =0 E = K + U 1 + U Poten<al energy associated with each force

18 Elastic Potential Energy Recall that the work done BY a spring on an object that moves from x 1 to x 2 is given by W = x 2 x 1 F s dx = 1 2 kx2 1 Is the spring force a conservative force? Yes, because the work only depends on the final and initial positions. Elastic potential energy: 1 2 kx2 2 Common to let U s (x=0) =0. Thus U s (x) = 1 2 kx2 U s = W = 1 2 kx kx2 1

19 Energy Diagram for Spring Energy diagrams help visualize the application of conservation of energy.

20 Chalkboard Question A 80-kg person is preparing to bungee jump off a bridge 90 meters above the ground with a 30 meter bungee of spring constant 40 N/m (the bungee acts like a spring once it starts stretching). Is the jump safe? How close does he get to touching the river?

21 Energy and Non-conservative Forces So far we have only looked at the energetics of systems involving conservative forces (and using potential energies associated with each force). What if there are also non-conservative force(s)? Recall the work-kinetic energy theorem expressed as W net = W i = K i U 1 U 2 + W NC = K W NC = K + U 1 + U E = W NC perhaps the most important equa<on of the semester!

22 JITT Comments how do we tell if the function can be represented by potential energy function I am having a hard time with knowing how and where to incorporate friction into the equations. I keep wanting to use the force of friction (for example mg(u)sin(theta)). I do not know if that is correct or not. How is friction accounted for in these types of problems?

23 Chalkboard Question A skier begins skiing down a hill of incline θ=45. The coefficient of kinetic friction is 0.2. If the skier started from rest, what is the skier s speed after it has traveled a distance L=100 meters?

24 Conservation of Energy You probably have heard that energy is conserved. But what about the equation E = W NC Does this equation imply energy isn t always conserved? Work done by non-conservative forces represents our ignorance or incomplete accounting of forms of energy. For example, friction causes the surface to heat up, increasing thermal energy. 1 st Law of Thermodynamics based on conservation of energy

25 Conservation of Energy Other forms of energy include electric fields (including chemical potential energy), magnetic fields, radiative energy, and rest-mass energy (E = mc 2 ). All fundamental forces of nature have been shown to be conservative. Conservation of energy is one of most trusted theorems of science!

26 Relationship between Force and Potential Energy Potential energy can be used to determine the properties of a particular conservative force. In 1 dimension, U = A B Fdx If B is very near A, say B = A + Δx, then U F x (F is approximately constant within this small region) Rearranging, and taking the limit of Δx getting very small: F = du dx

27 Clicker Question Q7.8 The graph shows the potential energy U for a particle that moves along the x-axis. At which of the labeled x-coordinates is there zero force on the particle? A. at x = a and x = c B. at x = b only C. at x = d only D. at x = b and d E. misleading question there is a force at all values of x

28 Clicker Question Q7.7 The graph shows the potential energy U for a particle that moves along the x-axis. The particle is initially at x = d and moves in the negative x- direction. At which of the labeled x-coordinates is the particle slowing down? A. at x = a B. at x = b C. at x = c D. at x = d E. more than one of the above

29 Equilibrium Locations

30 Chalkboard Question The potential energy of an object moving in one dimension is given by U(x) =A/x + Bx 2 For which values of x is the net force equal to zero?