!!
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 2nd half of this
lecture ◆ you may bring 1 8.5”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
!!
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=1/2mv2
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=1/2mv2
!!
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
!!
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
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
1 kg
!!
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
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 v2
l A hybrid car can convert some of that kinetic energy back into chemical energy stored in the battery
!!
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
!!
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
!!
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
J
maximum potential E mininum kinetic E
minimum potential E maximum kinetic E
!!
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?
l What units? ◆ J is a unit of energy ◆ So 1 J= 1 kg.m2/s2
◆ so h=20.4 m
l How fast is he going when he hits the bucket?
J
€
PE = mgh
h =PEmg
=10,000J
(50kg)(9.8m /s2)h = 20.4
€
KE =12mv 2 = ΔPE = 10000J
v 2 =(2)(10000J)50kg
= 400J /kg
v = 20m /s
!!
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
J
€
KE =12mv 2 = ΔPE = 5000J
v 2 =(2)(5000J)50kg
= 200J /kg
v = 14.1m /s
KE + PE = constant everywhere
!!
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 19th 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 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
!!
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
€
Efficiency =output energyinput energy
but your body is not so efficient at converting input energy to output energy ~10%
!!
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
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 =work done
time interval
!!
Power l We defined power as
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 kilowatt-hours ◆ power X time = unit of
energy
€
Power =work done
time interval
another illustrious Scotsman
1 kw-hr=1000 J/s * 3600 s = 3.6X106 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 = 6.72E6 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
!!
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/s2 (or N/kg)
◆ force – N(Newtons) ◆ mass – kg (kilograms)
l Understand the differences between science and pseudoscience
!!
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)
!!
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
Is there acceleration at A? At D? Are they the same?
Where is the acceleration negative?
!!
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 = x0 + v0xt
y = y0 + v0yt −
12gt 2
!!
Projectile Motion
l Assume that I release it 2 m from the ground
l yo=2m, voy=0 m/s
€
x = x0 + v0xt
y = y0 + v0yt −
12gt 2
€
y = yo −12gt 2
0 = 2m −12(9.83m /s2)t 2
t 2 =4m
9.83m /s2= 0.407s2
t = 0.64sx = xo + 20m /s(0.64s) = xo +12.8m
!!
Projectile Motion
l Suppose I throw it at 20 m/s at an angle of 45o
l Let’s again start with the vertical motion
l Now the horizontal motion
€
x = x0 + v0xt
y = y0 + v0yt −
12gt 2
€
0 = 2m + (20m /s)sin45o t − 12(9.83m /s2)t 2
0 = 2m + (20m /s)(0.707)t − (4.915m /s2)t 2
4.915t 2 −14.14t − 2 = 0t = 3.01s
€
x = x0 + (20m /s)cos45o tx = x0 + (20m /s)(0.707)(3.01s) = x0 + 42.6m
!!
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
!!
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
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
€
a =ΔvΔt
=20m /s− 0m /s
30s= 0.67m /s2
!!
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
!!
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)
!!
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 m1=mEarth=6X1024 kg l m2=myou=60 kg l d=Rearth=6.37X106 m
€
F = G m1m2
d2
F = 6.67X10−11Nm2 /kg2 (6X1024 kg)(60kg)
(6.37X106m)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
!!
Conservation laws
l Understand conservation of energy and conservation of momentum
l Be able to understand conversion between kinetic energy (KE=1/2mv2) and potential energy (mgh for gravity)
l Work = Force X Distance (units of Joules) l Work = ΔKE (or =ΔPE) l Power = Work/Time
!!
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 Prifle=Pbullet
!!
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?
l What units? ◆ J is a unit of energy ◆ So 1 J= 1 kgm/s2
◆ so h=20.4 m
l How fast is he going when he hits the bucket?
J
€
PE = mgh
h =PEmg
=10,000J
(50kg)(9.8m /s2)h = 20.4
€
KE =12mv 2 = ΔPE = 10000J
v 2 =(2)(10000J)50kg
= 400J /kg
v = 20m /s
