Download Objective: Conservation of Energy I

Type: Double
Date: _____________
Objective: Conservation of Energy I
Conservation of Energy II
Homework: Assignment (1-24)
Read 6.4, Do PROBS #’s (32, 37, 40, 41)
AP Physics “B”
Mr. Mirro
Date: ________
Conservation of Energy I
The gravitational force has an interesting property that when an object is moved from one place to
another, the WORK DONE by the gravitational force DOES NOT depend on the choice of PATH !
For instance, consider an object that moves from an initial height h0 to a final height hf along
three different paths.
Peak
A
B
C
The work done by gravity depends only on the initial and final heights and NOT on the paths between
these heights since,
PEg = m g Δh
For this reason, the gravitational force is called a conservative force, according to statement #1.
DefN: Statement #1 - A force is conservative when the work it does on a moving object is
independent of the path between the object’s initial and final position.
Another way to illustrate a conservative force is by considering a roller coaster track where the cart
moves through dips and double dips ultimately returning to its starting point. This kind of path is, which
begins and ends at the same place, is called a “closed” path.
“GRAVITY” provides the
only force that does work
on the car, assuming that
there is no friction or air
resistance.
Of course, the track exerts a NORMAL force, but this force is directed perpendicular to the motion and
does NO WORK. [Recall that cos 90° = 0]
On the downward parts of the trip, the gravitational force does positive work, increasing the car’s
kinetic energy and ultimately increasing its “speed”.
Conversely, on the upward parts of the motion, the gravitational force does negative work, decreasing
the car’s kinetic energy as well as decreasing its speed.
Over the entire trip, the gravitational forces does as much positive work as negative work, so the
“net work” is zero, and the car returns to its staring point with the same kinetic energy it had at
the start.
For this reason, the roller coaster satisfies another type of conservative force, this time - according to
statement #2.
DefN: Statement #2 - A force is conservative when it does no net work on an object moving
around a closed path, starting and finishing at the same point.
However, NOT ALL FORCES are conservative forces. A force is non-conservative if the work it does
on an object moving between two points depends on the path of the motion.
For instance, the consider the kinetic frictional force.
‰
When an object slides over a surface, the kinetic frictional force points “opposite”
to the sliding motion and does negative work.
‰
Between any two points, greater amounts of work are done over longer “paths”
between the points, so that the work due to friction depends on the choice of path.
‰
Thus, the kinetic “friction” force is a non-conservative force.
For a closed path, the total work done by a non-conservative force is NOT ZERO - as it is for a
conservative force.
For instance, a frictional force would oppose the motion and “slow” the car down.
Unlike gravity, friction would do negative work on the car through out the entire trip, on
both the up and down parts of the motion.
The truth is that MOST moving objects experience non-conservative forces, such as friction, air
resistance and propulsive forces, and the work Wnc done by the net external non-conservative forces
is NOT zero !
In these situations, the difference between the final and initial total mechanical energies is equal
to Wnc according to:
Wnc = Ef – Ei
Consequently, in theses instances - the total mechanical energy is not conserved.
AP Physics “B”
Mr. Mirro
Date: ________
Conservation of Energy I
Ex 1: A rescue helicopter lifts a 79 kg person straight up by means of a cable. The person has an upward
acceleration of 0.70 m/s2 and is lifted from rest through a distance of 11 m. [CutnellP6.20]
a. What is the tension in the cable ?
b. How much work is done by the tension in the cable ?
c. How much work is done by the person’s weight ?
d. Using the work-energy theorem, find the final speed of the person.
Ex 2: An extreme skier, starting from rest, coasts down a mountain that makes an angle of 25° wrt to the
horizontal. The coefficient of kinetic friction between her skis and the snow is 0.20. She coasts for
a distance of 10.4 m before coming to the edge of a cliff. Without slowing down, she skis off the
cliff and lands downhill at a point whose vertical distance is 3.5 m below the edge. How fast is she
going just before she leaves the cliff ? [CutnellP6.21mod]
Challenge: How fast is she moving just before she lands ??
AP Physlcs "B"
Mr. Mirro
Date: - - -
Conservation of Energy I
Ex 1: A rescue helicopter lifts a 79 kg person straight up by means of a cable. The person has an upward
acceleration of 0.70 m/s 2 and is lifted from rest through a distance of 11 m. [CutncllP .2 J
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Ex 2: An extreme skler, starting from rest, coasts down a mountain that makes an angle of 25° wrt to the
horizontal. The coefficient of kinetic friction between her skis and the snow is 0.20. She coasts for
a distance of 10.4 m before coming to the edge of a cliff. Without slowing down, she skis off the
cliff and lands downhill at a point whose vertical distance is 3.5 m below the edge. How fast is she
going just before she leaves the cliff? [Cutn IIP6._1 mod]
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Challenge: How fast is she moving just before she lands ??
AP Physics “B”
Mr. Mirro
Date: ________
Conservation of Energy II
While Newton’s laws provide a complete description of mechanical systems, applying Σ F = ma
directly isn’t always the easiest way to approach a problem. We can determine the properties of a system
directly from initial conditions, without knowing all the details of the motion between the initial and final
“states
“states.”
Knowing when to apply the Conservation Laws, when not to and what the conditions are
for conserving energy are essential when solving problems using an energy approach.
For example, suppose a mass is moved from ground level (point A) to a height (H) (point B). We can
calculate the work done by gravity for this process assuming that the mass is raised directly to the
final height.
B
E
Since the force due to gravity (Fg) acts opposite to the
displacement, the resulting work is:
Wgrav = - m g ΔH
Counter example, suppose the mass DOES NOT move directly from
the ground (A) to its final height (B), but follows the path ACDEB,
v
D
H
m
* since gravity does NO WORK along paths AC and EB,
the work done along CE is the “same” as the direct path.
* since the work done in this process is the same no matter
Fg = mg
which path you choose, energy is said to be “conserved.”
ground
A
C
Suppose a single conservative force (F) does work on a mass. The work-energy theorem
is WF = ΔKE. But because the force is conservative, you will have WF = - ΔU.
Therefore, combining these two concepts,
- ΔU = ΔKE
0 = ΔU + ΔKE ⇒ Δ (U + KE)
This means that the quantity K + U, the sum of the kinetic and potential energies, doesn’t change !
This known as the “Total Mechanical Energy.”
Where the Total Mechanical Energy (E) is expressed as:
E=K+U
If several conservative forces are acting, U is the sum of each potential energy, and if only
conservative forces are doing work on the system, you can apply the
law of conservation of mechanical energy:
E
initial
=E
final
As the system moves under the influence of conservative forces, the individual amount of kinetic and
potential energies change, but the sum of all of them stays the same.
Therefore, considering potential and kinetic energy yields the following:
Starting with the Conservation Principle,
E
initial
= E
final
PEinitial + KEinitial = PEfinal + KEfinal
Uinitial + Kinitial = Ufinal + Kfinal
mghi + ½ m vi 2 = mghf + ½ mvf 2
Now we can just consider the contribution of the initial and final states for each energy.
Either there is a presence or not – a binary condition on/off !
**************************************
Thus, rather than focusing on changing forces, speeds, positions, etc…the conservation
principle centers on what remains the same…namely, the energy of the system.
AP Physics “B”
Mr. Mirro
Date: ________
Conservation of Energy II
Ex 1: A motorcyclist is trying to leap across the canyon as shown by driving horizontally off the cliff at a
speed of 38 m/s. [Ignore air resistance and other non-conservative forces.] [Cutnell6.8]
a. Apply the Conservation of Energy Principle to
develop an expression in terms of PEg and KE.
b. Simplify the expression in part (a), and solve for the
speed of the cyclist, just as it touches down on hill #2,
in terms of g, h and v only.
c. Compute the speed with which the cycle strikes the ground on the other side.
d. Determine the vertical component of the motorcycle’s speed just due to free-fall and ΔH.
[Hint: Use a kinematic or free-fall approach]
e. Draw a free-body diagram representing the speed of the motorcycle at the instant it touches
down on hill #2 and use trigonometry to compute and confirm the motorcycle’s speed vf.
Ex 2: One of the fastest roller coasters in the world is the Magnum XL-200 at Cedar Point Park
in Sandusky, Ohio. The ride includes a vertical drop that accelerates the cart to a speed of
34 m/s [76 mi/hr] at the bottom of the dip. Assume that the coaster has a speed of nearly
zero as it crests the top to the hill. Neglecting friction and other non-conservative forces,
determine the approximate height of the peak. [Cutnell6.10sim]
[Hint: Use an energy approach]
Ex 2: One of the fastest roller coasters in the world is the Magnum XL-200 at Cedar Point Park
in Sandusky, Ohio. The ride includes a vertical drop that accelerates the cart to a speed of
34 m/s [76 mi/hr] at the bottom of the dip. Assume that the coaster has a speed of nearly
zero as it crests the top to the hill. Neglecting friction and other non-conservative forces,
determine the approximate height of the peak. [Cutnell6.10sim]
[Hint: Use an energy approach]