physics - giancoli 7e ch 06: work &...

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PHYSICS - GIANCOLI 7E

CH 06: WORK & ENERGY

INTRO TO ENERGY & ENERGY FORMS

● ENERGY: A physical quantity without a precise definition. We don't know exactly WHAT it is, but we know HOW it works.

- Energy "exists" in many forms; it cannot be created or destroyed, it’s always _________________ between types:

MECHANICAL (ME = _________) NON-MECHANICAL (NME)

POTENTIAL (U = ___________)

KINETIC

K = _________

THERMAL ENERGY

ALL OTHER TYPES

GRAVITATIONAL

UG = __________

ELASTIC/SPRING

UEL = __________

● The most important part about Energy is that it’s conserved. We use this to make old problems easier & solve new ones.

● Kinetic Energy (KE or K) has to do with an object’s ____________. An object that is moving has _____________ Energy.

● Potential Energy (PE or U) has to do with an object’s ________________.

- An object that is above the ground has ______________________ Potential Energy (____).

- A spring that is compressed or stretched (aka “deformed”) has ____________________ Potential Energy (_____).

● Mechanical Energy is simply the ________ of Kinetic & Potential. The unit of Energy is ___________ (___).

EXAMPLE 1: A 3-kg bird flies horizontally at 20 m above the ground with 10 m/s. Calculate its total Mechanical Energy.

EXAMPLE 2: You drop a 1-kg object from the top of a 100-m building. Find its Mechanical Energy:

(a) at the top of the building; (b) right before it hits the ground.

UAM EQUATIONS

(1) v = vo + a t

(2) v2 = vo2 + 2 a Δx

(3) Δx = vo t + ½ a t2

*(4) Δx = ½ ( vo + v ) t



WORK DONE BY A CONSTANT FORCE

● When you push against a box on a smooth horizontal surface:

- It begins to move, so it gains _______________ Energy. That Energy comes from _____________.

- This transfer of Energy from you to the box is called ____________. We say you “_______________” on the box.

The amount of WORK a force does on an object is the amount of Energy it gives to / takes from the object.

● The Work done by a constant force can be calculated using: (Since Work is Energy transferred, unit is _____)

WF = ________ = ______________ = ____________. - Where Θ is the angle between ____ and _____.

EXAMPLE 1: A 2-kg box is pulled horizontally by a 3-N force

for 5 m along a flat surface. Find the Work done by this force.

PRACTICE 1: You pull a 5-kg object vertically up with a

constant 100-N for 2 m. How much work do you do?

EXAMPLE 2: A 10-kg box on a smooth surface is pulled for 20 m, as shown below. (a) Find the Work done by F.

(b) Decompose F into FX and FY, and find the Work done by FX and FY.

PRACTICE 2: A water skier is pulled by a boat (shown). Both move north with a constant 10 m/s. The tension on the rope is

150 N, and makes an angle of 53o with the horizontal. How much work is done by the rope in 5 seconds?

10

F=50 N

20 m

37O

BOAT

53O



ZERO WORK, NEGATIVE WORK, AND WORK BY FRICTION

● Energy and Work are both Scalars, so they have no ________________.

- But depending on the direction of Force & displacement, the work done by the force can be ____ / ____ / ____:

- Positive Work: Force ___________________ motion (“_________” motion)

- Negative Work: Force ___________________ motion (“_________” motion)

- Zero Work: Force ___________________ motion (doesn’t affect motion)

EXAMPLE 1: A 6-kg box sits on a level surface. The coefficient of friction between the box and the surface is 0.7. You pull

horizontally on the box with 50 N for 8 meters. Calculate the work done by: (a) you, (b) friction, (c) weight, and (d) normal.

● The work done by Kinetic Friction is ALWAYS negative: W f k = ________________ = _________.

- Kinetic Friction dissipates Mech. Energy into ___________: MELOST = EDISS = _________.

● Some forces NEVER do work: Normal, Centripetal Forces, mg Y.

PRACTICE 1: A 3-kg box sits on a level surface. The box-surface coefficient of friction is 0.4. You pull on the box with 20 N

at 37o above the horizontal for 5 meters. Calculate the work done by: (a) you; (b) friction; (c) weight; and (d) normal.

BONUS: How much energy was dissipated? Where did this energy “go” ?

NOTE: Work is the Mechanical transfer of Energy. HEAT is the non-Mechanical transfer of Energy (due to diff in Temp).

WF = F Δx cosΘ

M



WORK DONE BY GRAVITY

● Remember, positive work “helps” motion, negative work “hurts” motion, and forces that don’t affect motion do NO work.

- When an object is FALLING, gravity does _____ work on it.

- When an object is GOING UP, gravity does _____ work on it.

● The force to lift something is: FLIFT = _________. (NOTE: Lifting implies from vi = ___ to vF = ___)

- So the WORK to lift is: WLIFT = __________________ = ___________ = ___________.

- Gravity does negative work: Wmg = __________________ = ___________ = ___________.

EXAMPLE 1: You lift a 3-kg box straight up from the floor and place it on a shelf 2 m-tall shelf above the floor. Calculate the

work on the object by: (a) you, (b) gravity. The box then falls from to the floor. (c) How much work is done by gravity then?

PRACTICE 1: You push on 3 kg box against a wall for a distance of 2 m with a 100-N force that makes 53o with the

horizontal, as shown. The box-wall coefficient of friction is 0.3. Calculate the work done by: (a) you, (b) friction, (c) gravity.

● The work done by gravity (ie. weight) depends only on the change in height, not the path. We call it path independent.

PRACTICE 1: A 70 kg person hikes from the bottom to the top of a 1,000 m hill with varying speeds. The path you take is

very irregular, with varying inclinations. How much total work does gravity do on you during the entire hike?

WF = F Δx cosΘ

53O



WORK ON INCLINED PLANES

● Remember Θ in W=F d cosΘ is the angle between Force and displacement.

- An inclined plane’s length, height, and angle are related by

EXAMPLE 1: A 100-kg crate slides at a constant 7 m/s from the top of a 12 m-long ramp that makes 37o with the horizontal.

(a) Find the crate’s initial grav. potential energy. (b) Use the Work Equation to calculate the work done by mgX, mgY, friction.

PRACTICE 1: You push a 2-kg box from the bottom of an inclined plane with 50 N for 10 m. The incline makes 37o with the

horizontal, and the box-incline coefficient of friction is 0.6. Find the work done by: (a) you, (b) friction, and (c) gravity.

EXAMPLE 2: A 5 kg box is pulled up at 3 m/s for a distance of 10 m, by a conveyor belt that makes 37o with the horizontal.

(a) How much work is done by gravity? (b) Which force is “pulling” the object up? (c) How much work does this force do?

Ug = mgh | WF = F Δx cosΘ



FS = –FA = ________

SPRING FORCE

● When you push/pull against a spring with FA, the spring pushes back (Newton’s _____ Law):

- x = _________________ (________________ or _______________).

- NOT the spring’s length, but its change x = ___________.

- k is the spring’s __________________________ (Units: _________)

- How __________ the spring is. Higher k __________ to deform.

- FS is a __________________ force, always opposite to deformation (____)

- Always pulling spring back to its original length (x = ____).

EXAMPLE 1: A 1.0 m-long spring is laid horizontally with one of its ends fixed. When you pull on it with 50 N, it stretches to

1.2 m. (a) What is the spring’s force constant? (b) How much force is needed to compress it to 0.7 m?

● If you attach a mass to a vertical spring, and let the mass come down slowly:

- Its weight will stretch the spring, until they reach _________________:

_______ = _______

- This also applies to a mass on top of a spring, slowly compressing it.

PRACTICE 1: A vertical spring is originally 60 cm long. When you attach a 5 kg object to it, the spring stretches to 70 cm.

(a) Find the force constant on the spring. (b) You now attach an additional 10 kg to the spring. Find its new length.



WF_____ = F______ Δx cosΘ

WORK DONE BY VARIABLE FORCES (AND SPRINGS)

● The work equation WF = F Δx cosΘ is only applicable for CONSTANT forces.

- For variable forces, we use the ________________ force instead

- The most important example is the spring force:

Graphically, the work done by ANY force is the ____________________ F-x.

EXAMPLE 1: You push a box along a level, smooth surface for 20 m. For the first 8 m, your force that grows steadily from 0 N to 60 N. For the last 12 m, your force decreases steadily from 60 N to 0 N. What total work do you do?

EXAMPLE 2: You hang a 5-kg object from a vertical spring. Once the system comes to its equilibrium, the spring has deformed by 10 cm. Calculate the work done by: (a) gravity; (b) the spring. (c) What is the spring’s potential energy?

PRACTICE 1: It takes 200 J of energy to compress a 1.0 m-long spring to 70 cm. How much work would it take to compress

this same spring from 70 cm to 50 cm?

FS = –kx | WF = F Δx cosΘ | UEL = ½ k x2

Wcompress = ________

WSpring = __________

F

x



CALCULATING NET WORK

● The NET or TOTAL work done on an object is simply the addition of ALL works done to the object (work by ALL forces).

WNET = _________ = ________________ OR W = F d cosΘ WNET = _________________.

- Unlike Forces, where we treat X & Y forces separately, work done by forces in BOTH X AND Y “form” NET work.

EXAMPLE 1: When a box is pulled by a constant force for a distance of 5 m up a smooth plane, the magnitude of the work

done by each force is as shown. Calculate: (a) the Net Work done on the box; (b) the Net Force acting on the box.

● To find the Net Work, you usually need to first identify all forces.

- But remember that NOT ALL forces do work.

PRACTICE 1: A 3-kg box is on a flat surface. The box-floor coefficient of friction is 0.6. When you pull horizontally on it for

10 m, it moves with 2 m/s2. Find the net work on the box. (Start by finding the magnitude of all forces acting on the box)

● If you don’t know all forces, but know the NET Force, you can still find the Net Work WNET = __________________.

PRACTICE 2: A 2-kg box is on a rough horizontal surface. When you pull horizontally on it, it moves with 3 m/s2. The

magnitude of your force and the box-floor coefficient of friction are unknown. What is the net work on the box across 5 m?

WF = F Δx cosΘ

WF = 75

Wmg = 60

WN = 0



THE WORK-ENERGY THEOREM

● The Work-Energy Theorem

- Gives us a way to find the initial or final velocity of an object that is acted upon by forces.

EXAMPLE 1: A 2-kg object has kinetic energy 4 J at

point A and speed 3 m/s at point B. Find its: (a) speed

at A; (b) kinetic energy at B; (c) net work from A to B.

PRACTICE 1: A 4-kg object has speed 6 m/s at point A, and

speed 10 m/s at point B. (a) How much work was done to it

between A and B? (b) If –32 J of total energy is done to the object

between B and C, what speed does it have at C?

● The Work-Energy Theorem is also very useful conceptually: If speed is constant ΔK = ____ WNET = ____.

EXAMPLE 2: A 5 kg box is on a rough horizontal

surface. The box-surface coefficient of friction is

0.6. You push on the box with a constant force, so

that the box moves with a constant 8 m/s. Find the

net work on the box.

PRACTICE 2: You lift a 3 kg object from the floor to a height of 2 m.

Find the: (a) work done by you; (b) work done by gravity; (c) net work

done on the object. You then walk horizontally with the object for 10

m. (d) How much work do you do?

WF = F Δx cosΘ

WNET = ΣW = WF,NET

K = ½ m v2

WNET = ΔK



PRACTICE: WORK-ENERGY THEOREM #1

EXAMPLE 1: A 5-kg block is initially at rest on a level, smooth surface. If the block is pushed with a

constant force of 20 N for a distance of 10 m, calculate its final speed.

EXAMPLE 2: How much work is needed to accelerate a 50-kg object from 3 m/s to 8 m/s?

PRACTICE 1: A 3-kg box is sliding on a smooth level surface with 8 m/s when it enters a 5 m rough patch. If the box has

half its original speed at the end of the patch, calculate the coefficient of kinetic friction between the box and the surface.

K = ½ m v2

WF = F Δx cosΘ

WNET = ΣW = WF,NET

WNET = ΔK




EXAMPLE 1: (a) How much work is needed to throw a 200-g ball with a horizontal 30 m/s? (b) If the

ball is in contact with your hand for 0.5 m while you throw it, what average force do you apply to it?

PRACTICE 1: Interstate roads in the U.S. typically have a speed limit of 70 mph (31.3 m/s). How much work is needed to

stop a 1,400 kg car moving at this speed? EXTRA: How much force is needed to accomplish this in 2 m?

PRACTICE 2: To prevent a crash, a driver slams on the brakes, causing his car to skid a total distance of 80 m before

stopping. The road-car coefficients of friction are 0.6 and 0.8. Calculate the car’s initial speed (the car’s mass is unknown).

K = ½ m v2

WF = F Δx cosΘ

WNET = ΣW = WF,NET

WNET = ΔK




EXAMPLE 1: A 8-kg box rests on a rough floor. The box-floor coefficient of friction is 0.5. Find the

work needed to move the box at a constant speed: (a) 3 m along the floor; (b) 3 m in the vertical.

PRACTICE 1: A 500-kg load is originally at rest on the floor. A crane pulls the load vertically up on the box with a constant

7,500 N until it reaches a height of 20 m. Calculate the speed of the load once it reaches 20 m.

PRACTICE 2: A 2-kg block is originally at rest at the bottom of a smooth inclined plane that makes 37o with the horizontal.

You push the block with a constant 20 N directed up the plane. Find the block’s speed after you push it 5 m up the plane.

K = ½ m v2

WF = F Δx cosΘ

WNET = ΣW = WF,NET

WNET = ΔK



CONCEPT: POWER ● Remember! Work is a change in _________________________ that results from a force being applied over a distance

- Since something is actively changing, a natural question to ask is “how FAST is it changing?”

- The answer to this question is __________________, which is simply the rate of change of an object’s energy

- The units of power are J/s, or WATTS, W EXAMPLE 1: A 150 g object moves in a straight line from (-2.5 cm, -4 cm) to (3 cm, -8 cm) under the influence of a 50 N force. At what rate is the object gaining energy? ● Consider the following special case: a car being propelled forward by some force F

- If the car experiences an equal force opposing it, it’ll move with constant velocity

- If the car moves a distance 𝑑, the work done by the engine is _____________

- If the car moves that distance in a time 𝑡, then the average power produced is ____________ ● Something that’s important to notice is that the force produced by the car isn’t changing; it’s in equilibrium

- Because it’s in equilibrium, it’s moving at a speed of _____________

- So, we see that, in the case of constant velocity motion, the power is equal to EXAMPLE 2: A 5 kg block slides down a 25o incline, held to a constant speed of 15 m/s by friction. If you measure the rate at which gravity changes the energy of the object to be 100 W, what is the coefficient of kinetic friction on the block?

● The AVERAGE POWER an object produces or consumes is

𝑃 = ____________ where 𝑊 is simply the amount of work done during the time Δ𝑡

𝐹ሬԦ𝑎𝑖𝑟

𝐹ሬԦ𝑐𝑎𝑟

𝑷𝒄𝒏𝒔𝒕 = ________



PRACTICE: POWER PRODUCED BY A FALLING OBJECT A 350 g mass can undergo two different motions, propelled by gravity in each case: it can either fall a height of 10 cm, or it can slide down a frictionless incline at a 30o angle. In which case is the mass’ kinetic energy changing faster?



ENERGY PROBLEMS WITH MULTIPLE OBJECTS

● When dealing with multiple objects, we still only use ONE equation, but must consider the energy of EACH object.

EXAMPLE 1: The system below is released from rest. If the 5-kg object is initially 3 m above the floor, calculate its speed

just before it hits the ground. (The string and pulley are massless, and you may disregard any effects due to friction)

EXAMPLE 2: The 3-kg block is 2 m above the floor, and the surface-block coefficient of friction for the 4-kg block is 0.5. If

the system is released from rest, find its speed just before the 3-kg block hits the floor. (The string and pulley are massless)

Ki + Ui + WNC = Kf + Uf WNC = WYOU + Wf

5 kg

4 kg

3 kg

4 kg



PRACTICE: ENERGY PROBLEMS WITH MULTIPLE OBJECTS

PRACTICE 1: The 4-kg block is 1 m above the floor, and the surface-block coefficient of friction for the 3-kg block is 0.4. If

the system is released from rest, find its speed just before the 4-kg block hits the floor. (The string and pulley are massless)

PRACTICE 2: The system below is released from rest. Calculate the speed of the system after the hanging block has

moved 1 meter. (The string and pulley are massless, and you may disregard any effects due to friction)


4 kg

3 kg

2 kg

30 kg

20 kg

53o



CONCEPT: ESCAPE VELOCITY ● In order to leave Earth, a rocket must be traveling at a minimum speed known as the escape velocity

- As a rocket moves away from the earth, gravity is pulling it back, slowing it down

- If the rocket doesn’t have enough initial speed, it won’t get far enough away before gravity saps all its speed ● In order to completely escape Earth’s gravity, the rocket must have __________ kinetic energy after it’s “escaped gravity”

- A rocket “escapes gravity” when the Earth has no more pull on it, or the potential energy due to gravity is zero

- This occurs infinitely far away from the Earth, since gravitational potential energy is inversely proportional to r

- We will say the limiting case is when the rocket has no kinetic energy ● This logic doesn’t have to be applied solely to a rocket leaving the Earth

- It can be applied to any object trying to leave any celestial body (a large mass in space, like a planet or moon)

- NASA had to know the escape velocity of the moon so the lunar lander could leave the surface

- Like everything we did with gravity, we are assuming the celestial body is spherical, which is why it has a radius EXAMPLE: The mass of the Earth is 5.97x1024 kg, and the radius is 6,371 km. What is the escape velocity of Earth?

● The ESCAPE VELOCITY for an object leaving the surface of a celestial body

𝑣𝑒𝑠𝑐 = ________________ where 𝑴 is the mass of the celestial body, and 𝑹 is the radius



PRACTICE: ESCAPING AN ORBIT AROUND MARS Astronauts are orbiting Mars at an altitude of 1,000 km and want to head home. How fast must they be going to leave their orbit around Mars? Note that the mass of Mars is 6.38x1023 kg, and the radius is 3,397 km.



m

m

OVERVIEW OF CONSERVATION OF ENERGY

● Conservation of Energy often refers to a _______________ (arbitrary group) of objects ESYS = E1 + E2 + …

● CONSERVATION #1: The TOTAL Energy of a System is conserved if the system is _____________________.

- A system is ___________________ IF NO ___________________ forces DO WORK on it (or IF ____________).

- Forces are either __________________ OR __________________, depending on how YOU defined the system.

EXAMPLE 1: A box slides on a rough floor. Assume kinetic friction is the only force doing work on it. For each of the choices

of a system below: (i) Is kinetic friction an internal force? (ii) Is the system isolated? (iii) Is the system’s energy conserved?

CHOICE OF SYSTEM (i) FRICTION INTERNAL?

(ii) SYSTEM ISOLATED?

(iii) ENERGY CONSERVED?

(a) System: Box only

(b) System: Box + Floor

● CONSERVATION #2: The MECHANICAL Energy of a System is conserved if the system is ___________________.

- A system is _________________ IF NO _______________________ forces DO WORK on it (or ____________).

- Forces are either Conservative (__________, ___________) OR Non-Conservative (___________, __________).

- Conservative Energies transfer, but system’s Mechanical Energy (ME) is conserved.

- Non-Conservative Add/remove Mechanical Energy to/from the system (ME is not conserved).

EXAMPLE 2: Each arrow below represents how energy in a system changes due to a different force. Indicate whether the

force corresponding to each arrow is (i) internal or external, and (ii) conservative or non-conservative.

SYSTEM (E = EA + EB) # Int/Ext C/NC

1

2

3

4

A

ME = U + K NME

POTENTIAL U= UG + UEL

KINETIC

K

THERMAL

& OTHERS

X

B

ME = U + K NME


KINETIC

K

THERMAL

& OTHERS

1 2 3 4



MORE CONSERVATION & *THE* CONSERVATION EQUATION

● The TOTAL Energy of a System is conserved if the system is ISOLATED (WEXT = 0).

- The MECHANICAL Energy of a System is conserved if the system is CONSERVATIVE (WNC = 0).

- Conservative Forces: Weight, Spring - Non-Conservative: Applied/You, Friction

So Mechanical Energy is conserved as long as no applied (“you”) or frictional forces DO WORK.

EXAMPLE 1: For each of the situations below: (a) Is the system isolated; conservative? (b) Describe the energy transfers.

SITUATION (a) SYSTEM ISOLATED?

(a) SYSTEM CONSERV?

(b) ENERGY TRANSFERS:

(a) A block is in free fall (no air resistance). We define the system

to be the block, the air, and the Earth.

(b)* A block that is falling feels the resistive force of air drag. We

define the system to be the block, the air, and the Earth.

(c) Assume a Styrofoam cup is sealed so no energy enters/leaves

the cup. The cup is at rest, but the coffee is heating the air inside

the cup. We define the system to be the cup and its contents.

NOTE: (1) The universe is an ISOLATED system; therefore the TOTAL Energy of the Universe is always conserved!

(2) If gravity does work (Δh), the Earth MUST be a part of the system weight is always an internal force!

● We will focus on Mechanical Energy. Combining equations for the Work done by Gravity & Spring and the Work-Energy

Theorem, we get the NEW, SEXY & RIDICULOUSLY useful Conservation* of [Mechanical] Energy Equation:

EXAMPLE 2: You release a 2 kg object from 100 m. Use Conservation of Energy to find its speed just as it hits the ground.

ME = U + K NME


KINETIC

K

THERMAL

& OTHERS



USING THE CONSERVATION OF ENERGY EQUATION

● Energy problems always include TWO or more points.

- The Conservation of Energy equation is very useful when the _________/_________ of an object is/are changing.

- The most important task is identifying which types of Energy / Work exist at the TWO points (initial & final).

EXAMPLE 1: You launch a 4-kg object directly up from the

ground with 40 m/s. Use Conservation of Energy to find the

maximum height the object will reach (ignore air resistance).

PRACTICE 1: You throw a 6-kg object down from a height

of 20 m. If the object reaches the ground with 30 m/s, what

initial speed did you throw it with? (ignore air resistance).

● In problems with more than TWO points, the TWO points you will pick will be the GIVEN & TARGET (known & unknown).

EXAMPLE 2: You launch an object directly up from the ground. The object is going up with 20 m/s at 30 m from the ground.

(a) Calculate its launch speed.

(b) Calculate its maximum height.


K = ½ m v2 U = Ug + UEL Ug = mgh



ENERGY EQUATION IN NON-CONSERVATIVE PROBLEMS

● If either friction or an applied / external force (“you”) does work on an object, Mechanical Energy will NOT be conserved.

- However, we can still use the Conservation* of Energy equation to solve these problems.

- Some of the simplest such problems are basic horizontal motion problems.

EXAMPLE 1: A 10-kg block is initially at rest on a level, smooth surface. If the block is pushed with a constant, horizontal

force of 20 N for a distance of 5 m, calculate its final speed.

EXAMPLE 2: A block of unknown mass is sliding on a flat surface with 30 m/s when it enters a long, rough patch. If the

coefficient of friction between the block and the rough patch is 0.6, calculate the block’s stopping distance.

● If you are asked to find the Work done by a force, you will typically use the Work Equation ( W = F d cosΘ ).

- But if you don’t have enough info for the Work Equation, the Conservation of Energy equation serves as a back-up!

EXAMPLE 3: You push a 5-kg block, initially at rest, along a rough, horizontal surface until it has reached 20 m/s. If kinetic

friction dissipates 200 J of energy in that interval, how much work did you do in pushing the block?


K = ½ m v2 WF = F d cosΘ Wf,k = – fk d



PRACTICE: NON-CONSERVATIVE PROBLEMS

PRACTICE 1: Your hand moves a horizontal distance of 1.6 meter while you throw a 0.140-kg baseball horizontally. If the

ball leaves your hand at 35 m/s, calculate: (a) the work done by you, and (b) the average force you exert on the ball.

PRACTICE 2: A 500-kg load is originally at rest on the floor. A crane pulls the load vertically up on the box with a constant

7,500 N until it reaches a height of 20 m. Calculate the speed of the load once it reaches 20 m.

PRACTICE 3: A 800-kg car leaves a skid mark of 90 m in stopping from 30 m/s. Calculate the car-road coefficient of friction.




ENERGY PROBLEMS WITH RESISTIVE FORCES (Linear)

● We can use the Conservation of Energy equation to solve problems with resistive forces, such as air/water resistance.

- Resistive forces work like kinetic friction: they always ________________ motion. W f, air = __________.

EXAMPLE 1: If a 2-kg object released from a height of 80 m reaches the floor with 30 m/s, calculate:

(a) the amount of work done by air resistance;

(b) the average force of air resistance.

PRACTICE 1: A 10-g bullet hits a wooden wall with a horizontal 300 m/s. If the bullet penetrates the wall by 5 cm, calculate:

(a) the amount of energy lost by the bullet.

(b) the average frictional force that stops the bullet.

EXAMPLE 2: A 70-kg diver jumps from a platform above the water. If the jumper enters the water with 10 m/s, and the

average resistive force of water on the diver is 2,000 N, how deep under water will he reach?




CONSERVATIVE FORCES ARE PATH INDEPENDENT

● Conservative forces have TWO key (and equivalent) properties:

(1) The Work done by Conservative forces does NOT depend on the ________________________.

- Since weight is a Conservative force, its work done depends only on the _______________ displacement.

So in ALL of the following, the final speed is the same (if frictionless):

(2) The NET Work done by Conservative forces is ZERO in a ________________________ (“round trip”):

- Weight is Conservative WNET = 0: - Friction is NOT Conservative WNET not 0:

EXAMPLE 1: A 2 kg block is released from the top of a smooth inclined plane that is 10 m long and makes 37o with the

horizontal. What speed will the block have once it reaches the bottom of the plane?

PRACTICE 1: A 3 kg block is initially moving on a flat surface when it reaches the bottom of an inclined plane with 20 m/s. If

the plane is smooth and makes 53o with the horizontal, how far up the plane will the block slide?


A B

C D A B

m



MORE INCLINED PLANE PROBLEMS

EXAMPLE 1: A 2-kg block is originally at rest at the bottom of a smooth inclined plane that makes 37o with the horizontal.

You push the block with a constant 20 N directed up the plane. Find the block’s speed after you push it 5 m up the plane.

EXAMPLE 2: A 3-kg crate is placed at 8 meters above the ground, on an inclined plane that makes 53o with the horizontal.

The block-surface coefficient of friction is 0.5. Calculate the block’s speed as it reaches the bottom of the plane.

EXAMPLE 3: What minimum speed does a 3-kg block need to have at the bottom of an inclined plane in order to make to

its top? The incline is 5 m long and makes 53o with the horizontal. The block-surface coefficient of friction is 0.4.




PRACTICE: INCLINED PLANE PROBLEMS

PRACTICE 1: When a 4-kg block is released from rest from the top of an inclined plane, it reaches the bottom with 4 m/s.

The incline is 5 m long and makes 37o with the horizontal. Calculate the magnitude of the frictional force acting on the block.

PRACTICE 2: When a block of unknown mass is released from the top of an inclined plane of length L meters, it slides

down. The incline makes an angle of Θ degrees with the horizontal, and the coefficient of kinetic friction between the block

and the plane is µ. Derive an expression for the speed of the block at the bottom of the plane.

PRACTICE 3: A block of unknown mass is released from a distance D1 from the bottom of an inclined plane, then slides on

a horizontal surface, and up a second inclined plane, as shown. Both planes make an angle of Θ degrees with the

horizontal. The horizontal surface is smooth, but the coefficient of friction between the block and the two inclines is µ.

Derive an expression for the maximum distance D2 that the block will reach on the second incline.




MOTION ALONG CURVED PATHS

● ALL curved path problems MUST be solved using Energy (there are no alternatives*).

EXAMPLE 1: What minimum speed would a block (of unknown mass) need to have at the bottom of the 30 m-high, smooth

hill shown below in order to reach its top?

EXAMPLE 2: A 70-kg skater dude rides on a half-pipe (a perfect semi-circle) of radius 3 m, shown. Calculate his speed:

(a) at B, if he starts at rest from A.

(b)* at A, to reach D, 2 m above C.

EXAMPLE 3: A cart goes around a loop-the-loop of radius R. Derive an expression for the minimum speed that the cart

needs to have at the bottom of the loop in order to reach the top of the loop, if the cart is locked to the tracks;


A

B

C

D



MORE ROLLERCOASTER PROBLEMS

EXAMPLE: A rollercoaster cart without seat belts goes around a loop-the-loop of radius R. Derive an expression for the:

(a) minimum speed needed at the TOP of the loop (point 3) so the cart makes it without passengers falling;

*(b) minimum speed needed at the BOTTOM of the loop (2) so the cart reaches the top (3) with the speed found in (a);

(c) minimum height H that the hill shown (point 1) must have so the cart reaches the top (3) with the speed found in (a);

*(d) force that the track will exert on the cart at the TOP of the loop (3) if it gets there with double the speed found in (a).


1

3 H=?

2



GRAV. POTENTIAL ENERGY IS RELATIVE + PENDULUMS

● In Energy problems, what matters is not an object’s initial or final height, but instead the __________________ in height.

- If you know an object’s change in height, you can set the “ground level” (________) to be at the LOWEST point*.

Ug is RELATIVE to an arbitrary (up to you) ______________________ height (“ground level”), where h = ____.

EXAMPLE 1: If the rollercoaster cart shown below moves passes point A with 20 m/s, calculate its speed at point B.

NOTE: If you do NOT set h = 0 at the lowest point in, points below h = 0 will have NEGATIVE Potential Energy.

● In most PENDULUM problems, you won’t know the distance between the floor and the pendulum’s lowest point.

EXAMPLE 2: A pendulum is built from a 2 kg bob and a 3 m-long light rope. It is attached to the ceiling, pulled until it is 1 m

above its lowest point, and released. Calculate: (a) the pendulum’s maximum speed; (b) the rope’s tension at the bottom.


A

B 10 m



PRACTICE: CURVED PATH PROBLEMS

PRACTICE 1: A 60 kg surfer is moving with 3 m/s at a certain point in a wave. Later on, he is moving with 8 m/s at a second

point, 2 meters lower. Calculate the work done by the wave on the surfer.




MORE PENDULUM PROBLEMS

● In some pendulum problems, you won’t be given heights, and will need to use the Pendulum Equation (which we’ll derive)

EXAMPLE 1: A pendulum is built from a 1 kg bob and a 2 m-long light rope. The pendulum is attached to the ceiling and

released from rest from 37o with the vertical. Calculate the pendulum’s maximum speed.

PRACTICE 1: A pendulum is built from a 3 kg bob and a 4 m-long light rope. It is attached to the ceiling and pulled from its

equilibrium position until it makes an angle of 53o with the vertical. It is then given an initial speed of 2 m/s directed down.

(a) Calculate the maximum speed that the pendulum will attain.

(b) Calculate the maximum angle that the pendulum will make with the vertical on the other side.




ENERGY PROBLEMS: BUMPS

EXAMPLE: A car rides over two consecutive bumps that are nearly circular with radii RA and RB (where RA > RB), as shown.

Derive an expression for the:

(a) maximum speed the cart can have at the top of the second bump (B) in order to NOT lose contact with the road;

*(b) maximum radius RA the first bump can have, in terms of RB, so the cart doesn’t lose contact with the road at B;

(c) force that the track will exert on the cart at the top of the second bump (B) if RA is 20% greater than RB.




PROJECTILE MOTION PROBLEMS WITH ENERGY

● Projectile Motion problems asking for speeds or heights are easier to solve using Conservation of Energy.

EXAMPLE 1: You throw a 5-kg object from the top of a 30 m-tall building with 20 m/s directed at an unknown angle above

the horizontal. The object reaches a maximum height of 40 m. Ignoring air resistance, calculate the object’s speed:

(a) at points B, C, and D;

(b) at D had it been thrown at any angle below the x-axis.

EXAMPLE 2: You throw a ball with 20 m/s at 37o above the horizontal. Use Conservation of Energy to find its max. height.

NOTE: Problems that require TIME to be solved cannot be solved with Energy; instead we use ___________________.


A

B

C

D



PRACTICE: PROJECTILE MOTION WITH ENERGY

PRACTICE 1: You are practicing jumping as far as you can. In one attempt, you run and leave the floor with 7 m/s directed

at an unknown angle. What maximum height do you reach if your speed at that point is 5 m/s? Ignore air resistance.

PRACTICE 2: When you launch a 3-kg object from the ground with unknown initial speed directed at 37o above the x-axis, it

hits the building shown below at 15 m above the ground with 25 m/s. Calculate the object’s launch speed.

PRACTICE 3: A 3-kg box is nudged off the top of the path shown below, slides down, and is launched form the lower end of

the path. The path is frictionless and its highest point is 10 m above the ground. The lower end is 2 m above the ground and

makes 53o with the horizontal. Calculate the box’s speed:

(a) at the lowest point in the path;

(b) just before it leaves the path;

(c) at its highest point;

(d) just before it hits the ground.




CONSERVATION OF ENERGY WITH SPRINGS: HORIZONTAL

● Problems involving springs and stationary objects are solved using _______________.

- Problems involving springs and moving objects are solved using _______________.

- Because Springs are always massless, they won’t have Kinetic (or Gravitational Potential) Energy, only Elastic.

EXAMPLE: A 4-kg block is on a smooth, horizontal surface. You push the block against a horizontal spring (shown below)

that is attached to a wall, with a constant, horizontal force. When this force is 100 N, the spring compresses 20 cm. When

you release the block, the spring pushes it out, accelerating the block to the right. Calculate:

(a) the spring’s force constant;

(b) the spring’s maximum potential energy;

(c) the block’s launch speed.

PRACTICE: A 4-kg block moving on a frictionless, horizontal surface with 20 m/s strikes a massless, horizontal spring of

force constant 600 N/m. Calculate the maximum distance that the block will compress the spring by .

EXTRA: What maximum acceleration will the block experience? (hint: this happens when it temporarily stops).

Ki + Ui + WNC = Kf + Uf FS = – k x UEL = ½ k x2

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ENERGY WITH HORIZONTAL SPRINGS AND FRICTION

● In problems with springs and rough surfaces, there are two forces and two distances: distance traveled and compression.

EXAMPLE: A massless spring of force constant 500 N/m is setup parallel to a long, horizontal surface, and attached to a

vertical wall, as shown below. A 3-kg block is then pushed against the spring until it is compressed by 0.4 m. The block-

surface coefficient of friction is 0.6. When you release the block, the spring pushes out, accelerating the block. Calculate:

(a) the speed with which the spring launches the block;

(b) how far the block will travel after it leaves the spring.

PRACTICE: A 4-kg block moving on a flat surface strikes a massless, horizontal spring of force constant 600 N/m with a 20

m/s. The block-surface coefficient of friction is 0.5. Calculate the maximum compression that the spring will experience.


m

m



ENERGY WITH SPRINGS: VERTICAL

● When a moving mass interacts with a vertical spring, we use ALL 3 types of Mechanical Energy: _____, _____, & _____.

EXAMPLE 1: A 5-kg block is released from 3 m above the top of a 400 N/m vertical spring. Calculate the maximum

compression that the spring will experience.

EXAMPLE 2: A vertical spring of force constant 800 N/m is attached to a horizontal surface, as shown below. A 10-kg block

is placed on top of the spring, and allowed to descent slowly, compressing the spring until the system reaches equilibrium.

The block is then pulled down an additional 30 cm, and released from rest. Calculate:

(a) the spring’s compression at equilibrium;

(b) the block’s maximum speed (where does this happen?);

(c) the block’s speed once the spring is back to its original height;

(d) the block’s maximum height (above the spring’s original height).


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physics - giancoli 7e ch 06: work &...

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