Common Exam 2 During class (1 - 2:55 pm) on 6/14, Tue Room: 412 FMH (classroom)
Bring scientific calculators Exam covers lltectures after common exam 1.
Review session: Monday during class
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Last class In 1D: Kinetic Energy Work Net Work Work-Energy Theorem
Today
Kinetic Energy, Work, Net Work, W-E Theorem in 2D
Conservative vs. Non-conservative forces Gravitational Potential Energy
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1 For 2D motion…
Kinetic energy
11222 KE..== mv m() vx + vy 22
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What if we have force and displacement with an angle? Work in 2D
2 Work in 2D What if force and displacement are perpendicular?
Example: Uniform circular motion
F Fnet net
Fnet
No change in “magnitude” of velocity No kinetic energy change No work, sorry ! (Velocity does change, because the “direction” changes.)
Displacement
positive work
Force
Displacement
zero work
Force
Displacement negative work
Force
Work vs. Disp.-Force angle Cos!
3 Work done by a constant force in 2D Define W = F d cos θ F , d ≡ F ⋅ d
Scalar (dot) product magnitude Force
θ Displacement
positive work Note: 090≤<θ θ = 90 zero work 90<≤θ 180 negative work 7
Example Angle = 20 degree, F = 5 N, d = 2 m. Find the work done by force F.
4 Math Review: Scalar (dot) product using components ddd= (, ) F = (,FFx y ) x y WFd=≡⋅cosθ Fd, Fd
=+Fdxx Fd yy Commutative property A⋅B=B⋅A Distributive property (A + B)⋅C = A⋅C + B ⋅C
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iClicker Quiz ForceFN=−(3 , 2 N ) is acting on an object, which makes a displacement dm= (2 , 4m)
Find the work done by this force.
A) (6JJ , - 8 )
22 22 B) 32+× 24 + C) 6 + 12 -4 -8 = 6 J D) -2 J E) Not enough information. I need to know theta.
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5 If multiple forces are applied on an object, F 1 vi v f F2 Displacement d F3 WFdFFF==+++ii( ...) d net net 123 =+++=+++Fd123iii Fd Fd... W 123 W W ... 11 =−=K K mv22 − mv fi22 f i
Work-Energy Theorem in 2D If multiple forces are applied on an object, F 1 vi v f F2 Displacement d F3
Wnet = W1 +W2 +... = K f − Ki
6 Example & iClicker
F=T = 10 N rope
2 kg d = 5 m θ = 30O
Frictionless surface
1. Find the forces on the box. A) PitiPositive B) Zero 2. Find the work done by each force. C) Negative 3. Find the net work 4. If initial velocity is 3 m/s, what is final kinetic energy?
Next topic
Conservative vs. Non-conservative forces Gravitational Potential Energy
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7 Motivation Why do we learn about potential energy and conservation of mechanical energy?
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iClicker Quiz Work done by a gravity force
hm1 = 4 Mass =1 kg Mass =1 kg hm1 = 4 hm= 3 + 3 vs. + - hm2 = 2 + Fmgg = hm4 =1 hm4 =1
(Work along h1->h2->h3->h4) vs (Work along h1->h4)
Which work is greater? A) Work along h1->h2->h3->h4 B) They are equal
C) Work along h1->h4 D) Not enough information 16
8 iClicker Quiz Work done by a gravity force x1=0 m x2=4 m hm1 = 3 Mass =1 kg
vs. Fmgg = vs.
hm4 = 0
((gWork along red arrows) vs (Work along gg)reen arrows) vs ((gwork along blue arrow)
Which work is the greatest? A) Work along green arrows B) Work along blue arrow C) Work along red arrows D) Work along red and green are equal and greater than work along blue arrow 17 E) They are all equal
Conservative force
If the work done by a force depends only on initial and final positions, the force is called a conservative force.
Gravity force is “a” conservative force.
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9 iClicker Q: Work done by friction force
Position 1m 2m 3m 4m
- - vs. fk =1N : opposite to - displacement x1 =1m x4 = 4m
- In which case, the work by friction is more negative? A) Top B) Bottom C) They are equal D) N.E.I. Is friction force a conservative force? 19
Non-Conservative force
If the work done by a force depends not only on initial and final positions, but also on the path between them, the force is called a non-conservative force.
Example: Friction force,Tension, normal force, and force applied by a person.
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10 Conservative forces in Phys 102: Gravity, Spring force(to be studied in future)
Non-Conservative forces in Phys 102: Friction, Normal force, Tension, Other applied forces
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Mass=1 kg Can you find the work done by height=10 m the gravity force along the path? A) Yes, of course. B) Please… No. C) Not enough information ☺
height=2 m
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11 Gravitational Potential Energy
(Work done by gravity on object with mass m ) hi = Wmghhg =−()if
Fmgg = = mggghmif− gh
=−UUgi,, g f h f where Umghg =
U g = Gravitational “Potential Energy", which depends on position only. Energy related to p osition, which p otentially converted to kinetic energy .
WUg =−gi,, U gf =−−() U gf , U gi , =−Δ U g Work done by gravity is negative of gravity Potential Energy change
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Gravitational Potential Energy Mass m
Gravitational Potential Energy: Height: h
Umghg =
(Note: Height, h, should be measured along vertical direction.)
The higher and the heavier an object is, the gggreater the gravitational p otential energy is.
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12 Example: Bungee Jump and Gravitational potential energy
(a)Is that, by any chance, …uh… Prof. Ahn? (b) Suppose that the person’s mass is 70 kg and the height of the bridge from the water surface is 120 m. What is the person’s gravitational potential energy when the person starts to jump? The height and the potential energy is measured from the water surface.
New Zealand, 2004
initial m= 3 kg
10 m
final
30 deg iClicker Quiz The change in gravitational potential energy is ______(a)3 x 9.8 x 10 J (b) 3 x 9.8 x (-10) J (c) something else
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13 initial m= 3 kg
10 m
final
30 deg Example: The change in gravitationagravitationall potential energy is ______J
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initial m= 3 kg
10 m
final
30 deg Example: The work done by gravity is ______
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14 WUg =−gi,, U gf =−−() U gf , U gi , =−Δ U g Work done by gravity force is negative of gravitational potential energy change
If gravitational force is the only force that does work, WWnet= g
, then , from Work-Energy theorem, WK net = Δ ,and above.
Δ=−ΔK U g Δ+Δ=KUg 0 KU+ g does not change.
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Conservation of Mechanical Energy with gravity force
Define Mechanical energy : 1 E ≡+KU =+mv2 mgh mech g 2
If WWnet= g or, if gravity force is the only forces that does work,
EEmech,, f= mech i : Mechanical energy does not change.
Mechanical energy is conserved.
In general, if conservative forces are the only forces that do work, mechanical energy is conserved.: “Conservation of mechanical energy”
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15 Conservation of Mechanical Energy If conservative forces only cause energy changes, the kinetic and potential energy
can change, but their sum, the mechanical energy Emec of the system, cannot change.
Example: Falling ball Less kinetic energy, but h MttilMore potential energy 1 v1 Height
MkitiMore kinetic energy, btbut Less potential energy h2 v2
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Conservation of Mechanical Energy If conservative forces only cause energy changes, the kinetic and potential energy
can change, but their sum, the mechanical energy Emec of the system, cannot change.
Example: Falling ball Mechanical energy h 1 1 v1 2 Emvmghmech =+ Height 2
Conservation of mechanical energy 11 mv22+= mgh mv + mgh h2 v2 2211 22
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16 iClicker Quiz A person throws a ball 30 degree from horizontal from the top of a 20 m high building. Neglect the air resistance. True or false? If he throws the ball at a different angle but with the same speed, the ball would hit the ground at a different speed. (a) True (b) False
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Example: Pendulum
2 m rope
30 degree V_i=0
What is the speed at the bottom?
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17 Example. A car with its engine off costs along a straight highway, which goes uphill. How far along the highway will the car go before its stops, if its initial speed was 80 km/h, and the slope is 15o? The tires roll, not skid.
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