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Momentum & Energy
Part A: Momentum & Impulse
Momentum
Momentum can be described as a characteristic of motion. It depends
on two things: mass and velocity. More specifically momentum is the
product of mass and velocity. Because it is the product of a scalar and
a vector……..
ρ = mv (units: kgm/s)
Impulse
Originally Newton did not consider his second law. In fact he
realized that the net force acting on an object is equal to the rate of
change of momentum.
ΣF = t
In other words, how much the momentum of an object changes
depends on two things: The force and the amount of time that the
force is applied.
Rearanging the above equation yields:
ΣFt = 
And the product of the force and this time is called “impulse” and is
taken into consideration when collisions take place. Its’ units are Ns.
Because the force during such collisions is rarely constant, the
average force is used. Or...if you are given a plot of force vs. time,
impulse is the area under the curve in that time interval (integral).
Practice
p. p.178 # 1 - 4
p. 192 RC # 1 - 4
Conservation of Momentum
If the net external force acting on an object or system of objects is
zero, then the total momentum of the object or system of objects
remains constant.
Another way of saying this is the momentum before is equal to the
momentum after an event. (if the net external force is zero)
mAvA + mBvB = mAvA’ + mBvB’
In fact it was the law of conservation of momentum which lead
Newton to his third law
The mass of a neutron was determined by having one collide with a
hydrogen nucleus and applying this law.
Samples
p. 185
Practice
p. 185 # 5 - 8
Practice
p. 192 RC # 1 - 10
p. 193 P # 1- 10
p. 188 # 9 - 12
Part B: Work, Power & Energy
Work
In one context is defined as the transfer of energy.
It can be manifested by
i)
change in speed of an object
ii)
change in height of an object
iii) change in shape of an object
Work is a scalar quantity with units
Joules(J) = Nm = kgm2/s2
In a mechanical sense, when a force (F) is applied to an object through
a distance (∆d), an amount of work is done given by the following
formula:
Work = Force x Displacement
Here the force and displacement vectors must be parallel. If they are
not,
i) take the component of the force parallel to the displacement and
multiply by the displacement or
ii) take the component of the displacement parallel to the force and
multiply by the force which ever is more convenient
In other words:
Example
W = F dcos
A man applies a force of 105 N to a lawnmower to keep it moving at a
constant velocity. The handle of the lawnmower forms a 30 degree
angle with the horizontal. If the lawnmower is pushed 50 m, how
much work has the man done? A: 4550 J
Samples
Practice
p. 199 # 1- 4
p. 202 # 5 - 8
Clarify “work done on” & “work done by”
Of course Force isn’t always constant
Work = area under a F vs. d curve
Practice
Illustration
Activity
p. 212 RC # 1 – 3
p. 212 AC # 1 – 5
p. 213 P # 1 – 5, & 7
Typical Energies (also see p. 221)
baseball of mass 250 g and going 25 m/s
60 W light bulb on for 1 hour
Human running for 1 Min
Radio for 1 Min
Heart for 1 Min
80 J
200,000 J
60,000 J
420 J
180 J
Is more work done lifting a crate or sliding it up an inclined plane?
See Inclined plane activity for gr. 10 ( in file cabinet)
Power - the rate at which energy is transferred (or the rate at which
work is done)
Power = ∆Energy / time = Work / time
Unit Watt (W) = J/s = Nm/s = kgm2/s3
Illustration Compare a 25 W light bulb and a 150 W light bulb
Compare my stereo to a good one
Compare my car to a good one
Example
If it takes the man (of the previous example) 12 seconds to push the
lawnmower 50 m, how much power does he exhibit? A: 380 W
Practice
p. 203 # 9 - 12
Activity
Calculate and compare the power of student in the class. Need - tape
measure, stop watch, and human scale (for mass)
Fact
SaskPower charges about 9 cents per kWhr (a unit for energy not
power)
1 kWhr is equivalent to the amount of energy “used” when 10, 100
Watt light bulbs are left on for 1 hour.
Question
How much does it cost to leave a block heater on over night? Assume
that overnight is 8 hours and P = 300W
A = 22 cents
Kinetic Energy - energy associated with motion
Potential Energy - stored energy - depends on position, shape or form
of an object
Derivation Consider a car accelerating from rest to a velocity v. To do this, a
force must be exerted over a distance (∆d) and so work must be done.
W = F∆d = mad
In fact the amount of work done is equal to the increase in the kinetic
energy of the car (if we neglect air resistance and so on).
W = ∆KE = KEf - KEi
KEf = mad
(but KEi = 0)
rearranging
vf2 = vi2 + 2ad for ad and subbing into previous expression
yields
KE = mv2/2
Example
A baseball with a mass of 180 g is thrown by a pitcher at a velocity of
27m/s.
a) What is the kinetic energy of the ball? A: 65.6 J
b)
Neglecting air resistance, how much work was done on the ball
to get it to move this fast?
A: 65.6 J
Sample
p. 221
Practice
p. 221 # 1 - 4
Potential Energy
Derivation Consider a box lifted several meters in the air. If the box is released it
will start to accelerate downward. There is work being done on the
box (a gravitational force is acting through a distance). Remember
that work is defined as a transfer of energy (from one form to
another). If this is the case there must be some form of energy from
which this kinetic energy is transferred. This form of energy is called
gravitational potential energy. The increase in potential energy can be
calculated if we know the amount of work that was done raising the
box.
W = ∆PE = F∆d = mg∆d or ∆PE = mgh
Note, this is not the potential energy but rather the change in
gravitational potential energy. There is therefore a need for a
reference position which we assign as zero potential energy (even
though it is surely non-zero).
Also, PE is path independant (it doesn't matter which path an object
takes when it is raised in a gravitational field...only the initial and
final positions).
Example
How much potential energy does a 30 kg object have when it is lifted
from the floor to a height of 2.0 m? A: 590 J
Sample
p. 224
Practice
p. 224 # 5 - 8
Conservation of Energy
Illustration Conservation of energy. Compare the kinetic and potential energy at
different points for a falling object, a roller coaster and a pendulum
(consider only the extremes to simpify )
Try This
Figuring physics - compare a ball travelling down two different tracks
in terms of time taken. (OHT - in class)
Figuring physics - compare three different projectile paths. Prove that
the speed is the same in each case. (OHT - in class)
Homework Derive an expression for the maximum speed of a pendulum bob.
Practice
p. 230 # 9 - 12