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Physics 3A: Energy
Shoup โ 118
So far we have explored motion using velocity, acceleration, forces^ Forces cause changes in motion => accelerations which arechanges in velocities. We can apply another approach to motion which in many cases ismuch simpler than forces, accelerations, etc.^ This uses the concept of
energy It won't tell us all the details, but is still very useful To use energy, we must separate the things we want to analyzefrom those outside that we don't want to analyze. Define the^
System^ as being the things we want to study
A single object or particle A collection of objects or particles A region of space It can vary in size and shape Define the^ environment
to be everything else outside the system^
Examples of systems and environments:^ Choice 1:^ system is ball^ environment is everything else^ the environment influences the systemby the tension in the rope & by gravity^ Choice 2:^ system is the ball, block, rope and pulley^ environment is everything else^ so tension in rope is now an internalforce, not external^ earth's gravity influences the system by external forces on boththe block and ball
Physics 3A: Energy
Shoup โ 120
Work^ We use the term work frequently in everyday life^ In physics we define work differently and more precisely^ Consider: You just ran out of gas and have to push your car to
the gas station. Case 1: You push your car with a force of 50 N for 100 m. Physics says you just did a significant amount of work
"on
the car" Case 2: You push and push on your car with a 50 N force but thecar never moves. Physics says you did zero work
"on the car"
, even
Physics 3A: Energy though you might have exerted a lot of effort
In physics we
define work
as (for constant forces): "The work done by an agent exerting a constant force on a systemis the product of the component Fcos(?) of the force along thedirection of the displacement of the point of application of theforce and the magnitude ?r of the displacement" in equation form: some properties of work:^ Dimensions?^ Units?^ Vector or scalar?^ if component of F in direction of displacement is zero, thenwork is zero.
W F r cos
(6.1) 22 ML/T
Physics 3A: Energy N m = Joule (J)scalar
ฮธ โ
Shoup โ 122
Question:^ Order the following situations in order of decreasingwork done on block, displacement is always to right:^
(^)
zero^ work
negative work positive work
Physics 3A: Energy
Is work done by (and if so, is it positive or negative):^ You on a heavy box as you lift it
Yes, positive
Gravity on the box while you lift it
Yes, negative
Gravity on the box as you carry it
No
You on the box as you carry it
No
You on the box as you lower it to the ground
Yes, negative
Gravity on the box as you lower it to the ground
Yes, positive
Frictional force between a car's tires and the road
No
on a car as it undergoes uniform circular motion
Physics 3A: Energy
Shoup โ 124
Work as a Scalar Product (dot product)^ Recall the scalar product of two vectors:^ From the previous definition of work, we can express it as the^ scalar product
of the two vectors
F^ and^ โ r^ as: ^ A B^ A B cos
^ ฮธ^
W F r cos
^ W Fr^
F r cos
ฮธ^ โ
Physics 3A: Energy
Some properties of scalar product^ commutative law:^ distributive law:^ unit vectors:^ Component form:^ Special case:
!##!"^ $"(6.5) A B^ B^ A^! #(&)+*^ '% A B^ C !#!(^ ' %%(6.6)A B^ A^ C^ ,,//^00 - .-^.^ -^. i i j j k^ k^1 (6.7) ,// 00 ,.^.^ .--- i j j k k^ i^0 (6.8) (^13542) A B^ A^ B^ Ax^ x^
(^5) B A^ By y z^ z^ (6.9) ^ A B^ A Bcos (^6629 278) A^ A^ A^ A^ xy
9 2 28 A^ A^ z
Physics 3A: Energy
Shoup โ 130
Lets compute the work done by spring on block as it movesfrom -xto 0max^
x^ f W F^ dxs s^ x^ i 0 W kx^ dxs x max 1 ^2 W k x^ s 2
0 x^ max
^ ^1 W k^0 s 2
^ ^12 ^ kx^ max^2
^12 ^2 k x^ max^2 12 ^ W^ k x^ s^ max^2
Physics 3A: Energy
For an arbitrary move from x
to x^ we get:i f^ What work do we do to stretch a spring?^ assume no acceleration^ then Fapp^
= - F^ s so
! ^!^ "# !^ #^ $##^ ^ "
x^ max^ % W F^ dxF app^ app 0 x max %^1 ^ W kx dx^ F app 2 0
(^2) k x max
x^ f^ &(') ^ W kx^ s x^ i
'^ **^12 dx k x^2
x^ f^ &^12 ''k x^ f^2 x^ i
) 12 k x i 2 1 W^ k xs^2
12 2 + k x^ if^ (6.15) 2
Physics 3A: Energy
Shoup โ 132
Lets look at how the work done on an object changes the object'smotion^ This approach will allow us to solve problems, easily whichwould have been hard using Newton's second law.^ Consider a block moving horizontally with a net horizontal force:^ if^ โx = x
-x^ then thefi^ work is: We can use Newton's 2
nd^ law to replace
, โ !ฮฃ - ฮฃF:
x^ f^ ^ W F dxnet x^ i . F m a
x^ f? W ma dxnet x^ i Physics 3A: Energy x^ f^ ^ ^ ^ W^ F dxnet^ x^ i
But a = dv/dt, so: We define ยฝ mv
2 as the kinetic energy of the block: x^ f? W ma dxnet x^ i
W^ net x^ fd v m dx^ d t x^ i
x^ fd v W m net d xx^ i d xdx^ d t
v^ fd x W m net d t v^ i
Wdv v^ f? m v dvnet v^ i
v^ f^ / 01 / 112 ^ m v 2 2 v^ i
12 22 m v m v^ fi 2
1 ^2 + W m v^ net f 2
Physics 3A: Energy^12 m v^ i^ (6.17)^2 132 K^ m v^ (6.18)^2
$%&'()*
So now we see that the net work done on an object goes intochanging the object's kinetic energy (K):^ W^ net This is called the work-kinetic energy theorem:^ "When work is done on a system and the only change in thesystem is in its speed, the work done by the net forceequals the change in kinetic energy of the system."^ Note: If work done is positive, kinetic energy increases (andso does speed). If work done is negative, kinetic energydecreases (and so does speed)^ Also note that this only involves scalars, not vectors!
K K K^ f i
12 3 K m v^ i i 2
123 K m v^ &f f 2
Physics 3A: Energy (6.19)
But wait! Is the work-kinetic energy theorem always valid?^ Consider book sliding across a table with friction.^ Frictional force does work on book, changing its kinetic energy^ But doesn't friction also do work on the table?^ Does the table's kinetic energy change?
Physics 3A: Energy
Shoup โ 136
Nonisolated Systems^ Any system which is "
influenced " by its environment Examples:^ Book pushed on a table^ Pencil picked up by you after you drop it Here the " influence
" causes a change in the system In our current context, the change in the system is its "
energy"
energy^ is either transferred into or out of the system^ This transfer of energy is the work done on the system We can expand our concept of the system's energy from justkinetic energy to include both
kinetic^ and
internal^ energy
Internal energy
- related to system's temperature (it is motion of system's "internal parts")
Physics 3A: Energy
So system's energy now is: For nonisolated systems, energy can betransferred into or out of the system by the environment^ Types of energy transfers are:^ Work^ Pushing your out-of-gas car^ Mechanical waves
- propagation of a disturbance in a mediasuch as air or water sound waves from your radio transferring energy into your ear Heat^ - transfer of internal energy Placing your hand on the campfire to see if it is hot Matter transfer - matter crosses the system boundary "fuel" into your car, or "exhaust" out of a rocket Electrical transfer - electrical current flowing from yourkitchen wall outlet into your finger Electromagnetic - radio waves received from local station
E= Internal Energyint^ K = Kinetic Energy
Physics 3A: Energy
E K Esystem int
Shoup โ 142
We have said work is the transfer of energy into or out of a system A logical question to ask is "
how fast or slow is the energy transferred?
This is the concept of Power:^ Power^ is the time rate of energy transfer^ The average power is defined as the work done divided by thetime interval (
โt) over which the work was done: Just as in other rate quantities, we can define the instantaneouspower by taking the limit:
W^ P ^ t W d W ^ (6.26)P lim t 0 t^ d t
Physics 3A: Energy
Shoup โ 143
We know from before, a small piece of work (dW) done over asmall displacement (d
r ) by a force
F^ is:
In general form, power is rate of transfer of any energy (E):^ Units of power are J/s, or watts (W): 1 W = 1 J/s = 1 kg m
23 /s
d W F d^ r
^ (6.27)P F^ v divide by dt
d W^ F d t^ ^ d r^ F^ v d t
d E 3 P (6.28)^ d t^ ^ 1 hp 550 ft lb^ s^ 746 W
Physics 3A: Energy
