Kinetic Energy. energy! l The kinetic energy of an object depends both on the mass of an object and its speed

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Emil Porter
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l 1 more day for LON-CAPA #4 l First exam: Feb 6 in Life Sciences A133 1:00 2:20 PM 40 questions, should not take full time review in 2 nd half of this lecture you may bring 1 8.

1 l 1 more day for LON-CAPA #4 l First exam: Feb 6 in Life Sciences A133 1:00 2:20 PM 40 questions, should not take full time review in 2 nd half of this lecture you may bring X11 sheet of paper with you (one-sided) hand-written; no xerox no form # needed for scoring sheet; no section no cellphones; evidence of a cellphone out will result in seizure of exam BPS Life Sciences

2 Kinetic Energy l If you push on an object, you can set it in motion l If an object is moving, then it is capable of doing work l It has energy of motion or kinetic energy l The kinetic energy of an object depends both on the mass of an object and its speed just like momentum l But in this case, the kinetic energy depends on the square of the speed KE=1/2mv 2 l and kinetic energy is a scalar quantity l The kinetic energy of a body is equal to the work required to bring it to that speed from rest net force X distance = kinetic energy Fd=1/2mv 2

3 Einstein s Big Idea l Emilie du Chatelet l Emilie had the insight that the kinetic energy of an object was proportional to the square of its speed l In an elastic collision, momentum is conserved, but kinetic energy is conserved as well

4 Gravitational potential energy l Work is required to elevate objects against Earth s gravity l The potential energy due to elevated positions is called gravitational potential energy l The amount of gravitational potential energy possessed by an elevated object is equal to the work done in moving it to its position l Suppose I have a 1kg ball 3 m above the ground l The work I have to do to lift the ball 3 m above the ground is W=Fd=(mg)h=mgh l This is the gravitational potential energy of the ball W=(1kg)(9.8N/kg)(3m)=29.4 J 1 kg Why mg as a force? Because gravity is pulling down on the ball with a force mg, and if I want to move it upwards at a constant velocity (no acceleration), then I must exert a force of mg in the opposite direction. Note that the potential energy is always defined with respect to some reference level, for example the ground or the floor of a building

5 Kinetic<->Potential l Potential energy can be turned into kinetic energy and vice versa l That s the whole fun of roller coasters

Work-Energy Theorem l When a car speeds up, its gain in kinetic energy comes from the work done on it l Or when a car slows down, work is done to reduce its kinetic energy l Work=ΔKE work equals the

6 Work-Energy Theorem l When a car speeds up, its gain in kinetic energy comes from the work done on it l Or when a car slows down, work is done to reduce its kinetic energy l Work=ΔKE work equals the change in kinetic energy applies to potential energy also l Since KE increases as the square of the speed, the work required to slow a car from a speed v to 0 goes as v 2 l A hybrid car can convert some of that kinetic energy back into chemical energy stored in the battery

7 Clicker question l Work is done on a car whenever it slows down l Suppose two cars of the same mass are travelling on the road car A is going four times as fast as car B both are braked to a complete stop l How much more work is done on car A than on car B? l A) the same work is done l B ) twice as much work l C) four times as much work l D) sixteen times as much work l E) sixty-four times as much work

8 Clicker question l Work is done on a car whenever it slows down l Suppose two cars of the same mass are travelling on the road car A is going four times as fast as car B both are braked to a complete stop l How much more work is done on car A than on car B? l A) the same work is done l B ) twice as much work l C) four times as much work l D) sixteen times as much work l E) sixty-four times as much work

9 Conservation of energy l Whenever energy is transformed or transferred, none is lost and none is gained l In the absence of work input or output, the total energy of a system remains constant l Consider the circus diver to the right l As he jumps off the platform, he loses PE but gains an equal amount of KE l Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes maximum potential E mininum kinetic E minimum potential E maximum kinetic E J

10 Conservation of energy l So the circus performer has a PE of 10,000 J (and a mass of 50kg) l How high up is he? PE = mgh h = PE mg = 10,000J (50kg)(9.8m /s 2 ) h = 20.4 l What units? J is a unit of energy So 1 J= 1 kg. m 2 /s 2 so h=20.4 m l How fast is he going when he hits the bucket? KE = 1 2 mv 2 = ΔPE = 10000J J v 2 = (2)(10000J) 50kg v = 20m /s = 400J /kg

11 Conservation of energy l How fast is he going when he s halfway down? l Half of the potential energy has been converted to kinetic energy KE = 1 2 mv 2 = ΔPE = 5000J J KE + PE = constant everywhere v 2 = (2)(5000J) 50kg v = 14.1m /s = 200J /kg

12 Joule s experiment l Where does the kinetic energy go when the guy hits the water? l This was a question that physicists had trouble with up to the 19 th century l They thought the energy disappeared l But it s transformed into heat

Joule s experiment l Joule proved this when he performed a very clever experiment l A falling weight causes paddles to turn inside a cylinder filled with water l The weight loses potential energy l

13 Joule s experiment l Joule proved this when he performed a very clever experiment l A falling weight causes paddles to turn inside a cylinder filled with water l The weight loses potential energy l Which is transformed into an increase in temperature of the water by the mechanical motion of the paddles l The heat energy gained by the water equaled the potential energy lost l And Joule gets a unit of energy named after him

14 Joule s experiment l Joule showed that 4200 J of work would always produce a 1 degree Centigrade temperature rise in 1 kg of water l This amount of energy is known as 1 Calorie (dietician) not to be confused with 1 calorie (physicist) which is the amount of energy needed to raise 1 g of water by 1 degree C l A 70 Calorie slice of bread has 70 Calories of stored chemical energy that can potentially provide 70 Calories of work and/ or thermal energy l 1000 calories = 1 Calorie but your body is not so efficient at converting input energy to output energy ~10% Efficiency = output energy input energy

15 Power l The definition of work says nothing about how long it takes to do the work l A measure of how fast the work is done is the power l Power is equal to the amount of work done per unit time Power = work done time interval l Power is also the rate at which energy is changed from one form to another l An engine with twice the power does not necessarily move a car twice as fast or twice as far, but it can do twice the work in the same amount of time it can produce a more powerful acceleration

Power l We defined power as Power = l So it must have units of energy over units of time (J/s) l We name the units of power (J/s) after James Watt, the developer of the steam engine in English units,

16 Power l We defined power as Power = l So it must have units of energy over units of time (J/s) l We name the units of power (J/s) after James Watt, the developer of the steam engine in English units, horsepower l Often will talk about kilowatts or megawatts l Note that your electric bill tells you not how many watts you used, but how many kilowatthours work done time interval power X time = unit of energy another illustrious Scotsman 1 kw-hr=1000 J/s * 3600 s = 3.6X10 6 J

A little calculation l BMR (basal metabolic rate) for an 18 year old (male) with a mass of 60 kg is ~1600 C(alories)/ day l This corresponds to 1600 C X 4200 J/C = 6.

17 A little calculation l BMR (basal metabolic rate) for an 18 year old (male) with a mass of 60 kg is ~1600 C(alories)/ day l This corresponds to 1600 C X 4200 J/C = 6.72E6 J l This is the energy consumed by your body just by sitting around (like in lecture) l There are 24X60X60=86,400 seconds in each day l Power = Energy/time l Power = 6.72E6 J/ 8.64 E4 s = 78 W l So just by sitting there, you re giving off as much heat as a 75 W light bulb 100 of you give off 7500 W

18 Basics l Understand scientific notation 0.003=3X10-3 =3E-03 understand km, cm, mm, µm, nm l Know the SI units for the quantities we have been working with position - m(meters) time - s (seconds) speed m/s acceleration m/s 2 (or N/kg) force N(Newtons) mass kg (kilograms) l Understand the differences between science and pseudoscience

19 Scalars, vectors and tensors l A scalar is a quantity that is just a number l A vector quantity has both a magnitude and a direction know how to add vectors l A tensor is a generalization of the above two quantities in multiple dimension rank 0 is a scalar rank 1 is a vector rank 2, or higher is a more complex quantity that has two directions and two magnitudes (curvature of space-time)

20 Motion l Know definition for displacement (position) distance=magnitude of displacement velocity speed=magnitude of the velocity acceleration l Be able to use a graph to determine position, velocity and acceleration at any point l Be able to use kinematic equations of motion for motion in x direction for vertical motion (with effects of gravity) projectile motion Where is the acceleration negative? Is there acceleration at A? At D? Are they the same?

21 Projectile Motion l Let s start simple l I throw the ball horizontally with a speed of 20 m/s l How long before it hits the ground? l How far has it travelled? x = x 0 + v 0 x t y = y 0 + v 0 y t 1 2 gt 2

22 Projectile Motion l Assume that I release it 2 m from the ground l y o =2m, v oy =0 m/s y = y o 1 2 gt 2 0 = 2m 1 2 (9.83m /s2 )t 2 4m t 2 = 9.83m /s = s2 t = 0.64s x = x o + 20m /s(0.64s) = x o +12.8m x = x 0 + v 0 x t y = y 0 + v 0 y t 1 2 gt 2

23 Projectile Motion l Suppose I throw it at 20 m/s at an angle of 45 o l Let s again start with the vertical motion 0 = 2m + (20m /s)sin45 o t 1 2 (9.83m /s2 )t 2 0 = 2m + (20m /s)(0.707)t (4.915m /s 2 )t t t 2 = 0 t = 3.01s l Now the horizontal motion x = x 0 + (20m /s)cos45 o t x = x 0 + (20m /s)(0.707)(3.01s) = x m x = x 0 + v 0 x t y = y 0 + v 0 y t 1 2 gt 2

24 Force, mass and laws of motion l A force is a push or pull l The mass of an object measures the amount of resistance to a change in motion or its inertia l Know the difference between Galileo s understanding of motion and that of Aristotle l Know Newton s 3 laws If the sum of forces is zero, the object will not accelerate F=ma For each force, there is an equal and opposite force l Know implications of Newton s 3 laws, for example, the Moon pulls as hard on the Earth as the Earth pulls on the Moon l Know how to carry out simple problems involving Newton s 3 laws l Impulse = Force X Time l Force is the rate of change of momentum. Given a graph of momentum vs time, you should be able to calculate the force

25 Force and acceleration l We had this problem in the homework l A car of mass 2180 kg slows down as the brakes are applied l What force is acting to slow the car down? l Note that the plot of speed vs time has a uniform slope a = Δv Δt = 20m /s 0m /s 30s = 0.67m /s 2 l If there s an acceleration then there must be a force causing that acceleration, and the force F=ma F = ma = (2180kg)(0.67m /s) = 1453N

26 Kepler s 3 laws and Newton s law of universal gravitation l Tycho Brahe detailed measurements that allowed Johannes Kepler to develop his 3 laws of planetary motion l Know Kepler s 3 laws and how Newton s law of gravity explains them elliptical orbits - mathematical result of Newton s law planets move faster when they are closer to the Sun -force of gravity is greater square of the period is proportional to the cube of the semi-major axis (radius) - farther away a planet is, the weaker gravity is

27 Newton s law of gravity l Know Newton s law of gravity and how to use it to calculate the force of gravity between two masses l Know why an astronaut in orbit appears weightless l Why do all objects fall at the same rate? Because the inertial mass (the mass in F=ma) is the same as the gravitational mass (the mass in the law of gravity)

28 Weight l Let s consider another force, your weight, i.e. the force the Earth exerts on you l Suppose you mass is 60 kg l m 1 =m Earth =6X10 24 kg l m 2 =m you =60 kg l d=r earth =6.37X10 6 m F = G m 1 m 2 d 2 F = 6.67X10 11 Nm 2 /kg 2 (6X10 24 kg)(60kg) (6.37X10 6 m) 2 F = 591N If you were twice as far away from the center of the Earth, your mass would be the same, but your weight would be 591N/4=148 N

29 Conservation laws l Understand conservation of energy and conservation of momentum l Be able to understand conversion between kinetic energy (KE=1/2mv 2 ) and potential energy (mgh for gravity) l Work = Force X Distance (units of Joules) l Work = ΔKE (or =ΔPE) l Power = Work/Time

30 Conservation of momentum l Let s go back to the rifle firing a bullet l Only an impulse external to the system can change the total momentum of a system l So the total momentum of the rifle + bullet system is conserved l So the momentum of the bullet equals the recoil momentum of the rifle Mv = mv l Since M >> m, V >> v l But P rifle =P bullet

31 Conservation of energy l So the circus performer has a PE of 10,000 J (and a mass of 50kg) l How high up is he? PE = mgh h = PE mg = 10,000J (50kg)(9.8m /s 2 ) h = 20.4 l What units? J is a unit of energy So 1 J= 1 kgm/s 2 so h=20.4 m l How fast is he going when he hits the bucket? KE = 1 2 mv 2 = ΔPE = 10000J J v 2 = (2)(10000J) 50kg v = 20m /s = 400J /kg

Gravitational potential energy

Gravitational potential energy Announcements l CAPA homework 4 due on Thursday Sept 27 at 10 AM l Please register your iclicker through LON-CAPA if you want to receive credit only a few of you have not l Help room hours (1248 BPS) Ian