Liceo Scientifico Isaac Newton Physics course Potential Energy and mechanical energy conservation Professor Serenella Iacino Read by Cinzia Cetraro

Potential energy Gravitational Potential energy Elastic potential energy

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

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 = 25000 from which m g = = 625 N (this is the weight). fig.3 AB A B A B A B 25000 40

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

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

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 →

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.

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

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

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.

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

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

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

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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