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**Liceo Scientifico Isaac Newton Physics course **

Potential Energy and mechanical energy conservation Professor Serenella Iacino Read by Cinzia Cetraro

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Potential energy Gravitational Potential energy Elastic potential energy

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**Gravitational potential energy represents the work done by gravity**

m → P → → h W = P ∙ s = P ∙ h ∙ cos 0° = mgh The Potential energy is indicated by the symbol U or only ( E P ). fig.1 A m → h B P W = m g h - m g h = U - U A A B A B B h fig.2

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**m g = = 625 N (this is the weight).**

Let’s make an example: A m → P h=40m B W = U - U m g h - m g h = U - U m g 40 – 0 = from which m g = = 625 N (this is the weight). fig.3 AB A B A B A B 25000 40

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**Gravitational potential energy depends only on the height h**

→ s the vertical route AB h → → → → θ W = P s= P AB = P h cos0°=mgh ∙ ∙ B AB C A → fig.4 s the route ACB h → → → → W = W + W = P AC + P CB = ∙ ∙ s → θ ACB AC CB B C = P AC cos(90°- )+ P CBcos90°= θ fig.5 h = mg = mgh sen θ sen θ

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**Conservative force → s → s → → → → v < - 1 2 m v W = < 0**

P → N → P → fig.6 v i f < - 1 2 m v W = < 0 from which v i f > - 1 2 m v W = > 0 from which

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**s → s → → → → → 1 2 v 1 m g h = m m g h = m v 2 - 1 2 m v = 0 W = 0 N**

A B P → P → 1 2 v A 2 1 m g h = m 2 fig.7 m g h = m v B 2 - 1 2 m v f i = 0 W = 0 P →

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**Elastic potential energy of a compressed spring**

U = 1 2 K x which represents the work done by the elastic force to pull the spring back towards its original length. We can observe that the work depends only on the compression x and so on the initial and final positions of the spring, therefore the elastic force is a conservative force. However not all forces are conservative.

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**Non conservative forces: Friction**

→ → F a → F a A B s → s → fig.8 → F a D C → F a s → W = W + W + W + W = = - F a s - F a s - F a s - F a s - 4 F a s AB BC CD DA

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**E = U + K E E = Mechanical Energy**

It is conserved only in systems where conservative forces are involved. 1 2 1 2 m v f 2 - m v i 2 the work – energy theorem: W = = K - K sum f i the difference in potential energy: W = U - U i f conservative force K - K f i = U - U i f from which we have U + K = U + K f i i E = E initial final

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**highest point - highest gravitational potential energy**

If there is no friction, the Roller Coaster is a demonstration of Energy Conservation. highest point - highest gravitational potential energy fig.9 Mechanical energy remains constant.

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**Spring and energy conservation**

→ When the object compresses the spring, its kinetic energy decreases and is transformed into elastic potential energy. m fig.10 v → When the motion is reversed, the potential energy decreases while the kinetic energy increases and when the object leaves the spring, the kinetic energy returns to its initial value. m fig.11

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**The pinball machine: → →**

s → To fire the ball of mass m, suppose we compresse the spring, having a constant equal to K, by length x. Ignoring friction, we want to know what is the launch velocity of the ball. P → fig.12 1 2 1 2 2 2 U + K f i = K x + 0 = 0 + m v f 2 K x v = m s f m

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**two children, two slides, no friction, same height h,**

Water Park: two children, two slides, no friction, same height h, h h v 1 v 2 U + K f i = 1 2 1 2 2 2 m g h + 0 = 0 + m v m g h + 0 = 0 + m v 1 2 from which v = 2 g h and v = 2 g h 1 2

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**Law of energy conservation is no longer valid.**

Conservative and non conservative forces: W = W cons + W non cons sum W cons W non cons = K - K f i + W = K - K f i sum W U - U i f U - U i f W non cons + = K - K f i cons = W non cons = U + K f - U + K i W non cons = E final - E initial Law of energy conservation is no longer valid.

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energy THE END

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