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

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

P → m A B h=40m fig.3 Let’s 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

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

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.

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

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

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

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.

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