Reading and Review

A mass attached to a vertical spring causes the spring to stretch and the mass to move downwards. What can you say about the spring’s potential energy (PE s ) and the gravitational potential energy (PE g ) of the mass? a) both PE s and PE g decrease b) PE s increases and PE g decreases c) both PE s and PE g increase d) PE s decreases and PE g increases e) PE s increases and PE g is constant Question 8.5 Springs and Gravity

A mass attached to a vertical spring causes the spring to stretch and the mass to move downwards. What can you say about the spring’s potential energy (PE s ) and the gravitational potential energy (PE g ) of the mass? a) both PE s and PE g decrease b) PE s increases and PE g decreases c) both PE s and PE g increase d) PE s decreases and PE g increases e) PE s increases and PE g is constant The spring is stretched, so its elastic PE increases, because PE s = kx 2. The mass moves down to a lower position, so its gravitational PE decreases, because PE g = mgh. Question 8.5 Springs and Gravity

Question 8.9 Cart on a Hill A cart starting from rest rolls down a hill and at the bottom has a speed of 4 m/s. If the cart were given an initial push, so its initial speed at the top of the hill was 3 m/s, what would be its speed at the bottom? a) 4 m/s b) 5 m/s c) 6 m/s d) 7 m/s e) 25 m/s

Question 8.9 Cart on a Hill When starting from rest, the cart’s PE is changed into KE:  PE =  KE = m(4) 2 A cart starting from rest rolls down a hill and at the bottom has a speed of 4 m/s. If the cart were given an initial push, so its initial speed at the top of the hill was 3 m/s, what would be its speed at the bottom? a) 4 m/s b) 5 m/s c) 6 m/s d) 7 m/s e) 25 m/s When starting from 3 m/s, the final KE is: KE f = KE i +  KE = m(3) 2 + m(4) 2 = m(25) = m(5) 2 Speed is not the same as kinetic energy

8-4 Work Done by Nonconservative Forces In the presence of nonconservative forces, the total mechanical energy is not conserved: Solving, (8-9)

8-4 Work Done by Nonconservative Forces In this example, the nonconservative force is water resistance:

You see a leaf falling to the ground with constant speed. When you first notice it, the leaf has initial total energy PE i + KE i. You watch the leaf until just before it hits the ground, at which point it has final total energy PE f + KE f. How do these total energies compare? a) PE i + KE i > PE f + KE f b) PE i + KE i = PE f + KE f c) PE i + KE i < PE f + KE f d) impossible to tell from the information provided Question 8.10a Falling Leaves

You see a leaf falling to the ground with constant speed. When you first notice it, the leaf has initial total energy PE i + KE i. You watch the leaf until just before it hits the ground, at which point it has final total energy PE f + KE f. How do these total energies compare? a) PE i + KE i > PE f + KE f b) PE i + KE i = PE f + KE f c) PE i + KE i < PE f + KE f d) impossible to tell from the information provided As the leaf falls, air resistance exerts a force on it opposite to its direction of motion. This force does negative work, which prevents the leaf from accelerating. This frictional force is a nonconservative force, so the leaf loses energy as it falls, and its final total energy is less than its initial total energy. Question 8.10a Falling Leaves Follow-up: What happens to leaf’s KE as it falls? What net work is done?

8-5 Potential Energy Curves and Equipotentials The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy:

8-5 Potential Energy Curves and Equipotentials The potential energy curve for a spring:

8-5 Potential Energy Curves and Equipotentials Contour maps are also a form of potential energy curve:

Lecture 11 Linear Momentum

Momentum is a vector; its direction is the same as the direction of the velocity.

Going Bowling I p p a) the bowling ball b) same time for both c) the Ping-Pong ball d) impossible to say A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. Which one of the two has the greater kinetic energy?

Going Bowling I p p a) the bowling ball b) same time for both c) the Ping-Pong ball d) impossible to say A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. Which one of the two has the greater kinetic energy? Momentum is p = mv so the ping-pong ball must have a much greater velocity Kinetic Energy is KE = 1/2 mv 2 so (for a single object): KE = p 2 / 2m

Momentum and Newton’s Second Law Newton’s second law, as we wrote it before: is only valid for objects that have constant mass. Here is a more general form, also useful when the mass is changing:

Change in Momentum Change in momentum: (a) mv (b) 2mv

A net force of 200 N acts on a 100-kg boulder, and a force of the same magnitude acts on a 130-g pebble. How does the rate of change of the boulder’s momentum compare to the rate of change of the pebble’s momentum? a) greater than b) less than c) equal to Momentum and Force

A net force of 200 N acts on a 100-kg boulder, and a force of the same magnitude acts on a 130-g pebble. How does the rate of change of the boulder’s momentum compare to the rate of change of the pebble’s momentum? a) greater than b) less than c) equal to The rate of change of momentum is, in fact, the force. Remember that F =  p/  t. Because the force exerted on the boulder and the pebble is the same, then the rate of change of momentum is the same. Momentum and Force

Impulse Impulse is a vector, in the same direction as the average force. The same change in momentum may be produced by a large force acting for a short time, or by a smaller force acting for a longer time. Impulse quantifies the overall change in momentum

Impulse We can rewrite as So we see that The impulse is equal to the change in momentum.

Why we don’t dive into concrete The same change in momentum may be produced by a large force acting for a short time, or by a smaller force acting for a longer time.

Going Bowling II Going Bowling II p p a) the bowling ball b) same time for both c) the Ping-Pong ball d) impossible to say A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest?

Going Bowling II Going Bowling II We know: Here, F and  p are the same for both balls! It will take the same amount of time to stop them. p p so  p = F av  t a) the bowling ball b) same time for both c) the Ping-Pong ball d) impossible to say A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest? av  t  p F 

Going Bowling III Going Bowling III p p A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, for which is the stopping distance greater? a) the bowling ball b) same distance for both c) the Ping-Pong ball d) impossible to say

Going Bowling III Going Bowling III p p Use the work-energy theorem: W =  KE. The ball with less mass has the greater speed, and thus the greater KE. In order to remove that KE, work must be done, where W = Fd. Because the force is the same in both cases, the distance needed to stop the less massive ball must be bigger. A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, for which is the stopping distance greater? a) the bowling ball b) same distance for both c) the Ping-Pong ball d) impossible to say

Conservation of Linear Momentum The net force acting on an object is the rate of change of its momentum: If the net force is zero, the momentum does not change! A vector equation Works for each coordinate separately With no net force:

Internal Versus External Forces Internal forces act between objects within the system. As with all forces, they occur in action-reaction pairs. As all pairs act between objects in the system, the internal forces always sum to zero: Therefore, the net force acting on a system is the sum of the external forces acting on it.

Momentum of components of a system Internal forces cannot change the momentum of a system. However, the momenta of components of the system may change. An example of internal forces moving components of a system: With no net external force:

Kinetic Energy of a System Another example of internal forces moving components of a system: The initial momentum equals the final (total) momentum. But the final Kinetic Energy is very large

Opposite case: Two identical cars travelling at identical speeds in opposite directions collide head on. BUT: VERY inelastic collision!

Nuclear Fission I Nuclear Fission I A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the greater momentum? a) the heavy one b) the light one c) both have the same momentum d) impossible to say 1 2

Nuclear Fission I Nuclear Fission I A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the greater momentum? a) the heavy one b) the light one c) both have the same momentum d) impossible to say 1 2 The initial momentum of the uranium was zero, so the final total momentum of the two fragments must also be zero. Thus the individual momenta are equal in magnitude and opposite in direction.

Nuclear Fission II Nuclear Fission II a) the heavy one b) the light one c) both have the same speed d) impossible to say 1 2 A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the greater speed?

Nuclear Fission II Nuclear Fission II We have already seen that the individual momenta are equal and opposite. In order to keep the magnitude of momentum mv the same, the heavy fragment has the lower speed and the light fragment has the greater speed. a) the heavy one b) the light one c) both have the same speed d) impossible to say 1 2 A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the greater speed?

Systems with Changing Mass: Rocket Propulsion If a mass of fuel Δm is ejected from a rocket with speed v, the change in momentum of the rocket is: The force, or thrust, is

A plate drops onto a smooth floor and shatters into three pieces of equal mass. Two of the pieces go off with equal speeds v along the floor, but at right angles to one another. Find the speed and direction of the third piece. We know that p x =0, p y = 0 in initial state and no external forces act in the horizontal

An 85-kg lumberjack stands at one end of a 380-kg floating log, as shown in the figure. Both the log and the lumberjack are at rest initially. (a) If the lumberjack now trots toward the other end of the log with a speed of 2.7 m/s relative to the log, what is the lumberjack’s speed relative to the shore? Ignore friction between the log and the water. (b) If the mass of the log had been greater, would the lumberjack’s speed relative to the shore be greater than, less than, or the same as in part (a)? Explain.