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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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1. Gravitational Potential energy 2. Elastic potential energy Potential energy

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

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P m A B h=40m fig.3 Lets make an example: 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). BA AB BA B A

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Gravitational potential energy depends only on the height h h C B A θ s s h C B A θ s the route ACB the vertical route AB W = P s= P AB = P h cos0°=mgh AB = P AC cos(90°- )+ P CBcos90°= h sen θ θ = mg = mgh W = W + W = P AC + P CB = ACBACCB θ fig.5 fig.4

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N N P P s s Conservative force v i f v > m v f i 2 W = > 0from which v i f v < m v f i 2 W = < 0 from which fig.6

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m g h = 1 2 v A 2 m 1 2 v B 2 m N P s N P s AB m v f i 2 = 0 P W = 0 fig.7

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Elastic potential energy of a compressed spring U = 1 2 K x 2 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: fig.8 s s s s DC A B F a F a F a F a W = W + W + W + W = = DA CDBCAB - F a s a s a s - 4 F a s - F a s Friction

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Mechanical Energy E = U + K the work – energy theorem: sum = m v f i 2 W = K - K fi the difference in potential energy: It is conserved only in systems where conservative forces are involved. W conservative force = U - U if if K - K fi =U + K ff ii =from which we have E initial = E 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. Mechanical energy remains constant. fig.9

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Spring and energy conservation m v fig.10 v fig.11 When the object compresses the spring, its kinetic energy decreases and is transformed into elastic potential energy. m 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.

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The pinball machine: s fig.12 N P 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. U + K ff ii = 1 2 K x v f 2 m+ 0 = 0 + v f = m 2 K x m s

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

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Conservative and non conservative forces: sum =WW cons W non cons + sum W K - K fi = fi = W cons W non cons + W cons = U - U if W non cons + U - U if K - K fi = W non cons U + K ff ii =- E initial - E final W non cons = Law of energy conservation is no longer valid.

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THE END energy

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