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Understanding Elastic Potential Energy and Simple Harmonic Motion

Learn about Hooke's Law, elastic potential energy, Simple Harmonic Motion, and the conservation of energy involving springs, gravity, and pendula. Discover how to calculate potential energy and the relationship between kinetic and potential energies.

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Understanding Elastic Potential Energy and Simple Harmonic Motion

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  1. Springs And pendula, and energy

  2. Elastic Potential Energy • What is it? • Energy that is stored in elastic materials as a result of their stretching. • Where is it found? • Rubber bands • Bungee cords • Trampolines • Springs • Bow and Arrow • Guitar string • Tennis Racquet

  3. Hooke’s Law • A spring can be stretched or compressed with a force. • The force by which a spring is compressed or stretched is proportional to the magnitude of the displacement (Fa x). • Hooke’s Law: Felastic = -kx Where: k = spring constant = stiffness of spring (N/m) x = displacement

  4. Hooke’s Law Felastic = -kx k = spring constant = 10 (N/m) x = displacement = 0.2m F = - (0.2m)(10 N/m) = -2N Why negative? Because the direction of the Force and the displacement are in opposite directions.

  5. Hooke’s Law – Energy • When a spring is stretched or compressed, energy is stored. • The energy is related to the distance through which the force acts. • In a spring, the energy is stored in the bonds between the atoms of the metal.

  6. Hooke’s Law – Energy • F = kx • W = Fd • W = (average F)d = d(average F) • W = d*[F(final) – F(initial)]/2 • W = x[kx - 0 ]/2 • W = ½ kx2 = D PE + D KE

  7. Hooke’s Law – Energy • This stored energy is called Potential Energy and can be calculated by PEelastic = ½ kx2 Where: k = spring constant = stiffness of spring (N/m) x = displacement • The other form of energy of immediate interest is gravitational potential energy • PEg = mgh • And, for completeness, we have • Kinetic Energy KE = 1/2mv2

  8. Simple Harmonic Motion & Springs • Simple Harmonic Motion: • An oscillation around an equilibrium position in which a restoring force is proportional the the displacement. • For a spring, the restoring force F = -kx. • The spring is at equilibrium when it is at its relaxed length. • Otherwise, when in tension or compression, a restoring force will exist.

  9. Restoring Forces and Simple Harmonic Motion • Simple Harmonic Motion • A motion in which the system repeats itself driven by a restoring force • Springs • Gravity • Pressure

  10. Harmonic Motion • Pendula and springs are examples of things that go through simple harmonic motion. • Simple harmonic motion always contains a “restoring” force that is directed towards the center.

  11. Simple Harmonic Motion & Springs • At maximum displacement (+ x): • The Elastic Potential Energy will be at a maximum • The force will be at a maximum. • The acceleration will be at a maximum. • At equilibrium (x = 0): • The Elastic Potential Energy will be zero • Velocity will be at a maximum. • Kinetic Energy will be at a maximum

  12. Simple Harmonic Motion & Springs

  13. The Pendulum • Like a spring, pendula go through simple harmonic motion as follows. T = 2π√l/g Where: • T = period • l = length of pendulum string • g = acceleration of gravity • Note: • This formula is true for only small angles of θ. • The period of a pendulum is independent of its mass.

  14. Example 3 Changing the Mass of a Simple Harmonic Oscilator 10.3 Energy and Simple Harmonic Motion A 0.20-kg ball is attached to a vertical spring. The spring constant is 28 N/m. When released from rest, how far does the ball fall before being brought to a momentary stop by the spring? What about a 0.4 kg ball?

  15. Simple Harmonic Motion & Pendula • At maximum displacement (+ y): • The Gravitational Potential Energy will be at a maximum. • The acceleration will be at a maximum. • At equilibrium (y = 0): • The Gravitational Potential Energy will be zero • Velocity will be at a maximum. • Kinetic Energy will be at a maximum

  16. Conservation of Energy & The Pendulum • (mechanical) Potential Energy is stored force acting through a distance • If I lift an object, I increase its energy • Gravitational potential energy • We say “potential” because I don’t have to drop the rock off the cliff • Peg = Fg * h = mgh

  17. Conservation of Energy • Consider a system where a ball attached to a spring is let go. How does the KE and PE change as it moves? • Let the ball have a 2Kg mass • Let the spring constant be 5N/m

  18. Conservation of Energy • What is the equilibrium position of the ball? • How far will it fall before being pulled • Back up by the spring?

  19. Conservation of Energy & The Pendulum • (mechanical) Potential Energy is stored force acting through a distance • Work is force acting through a distance • If work is done, there is a change in potential or kinetic energy • We perform work when we lift an object, or compress a spring, or accelerate a mass

  20. Conservation of Energy & The Pendulum Does this make sense? Would you expect energy to be made up of these elements? • Peg = Fg * h = mgh • What are the units?

  21. Conservation of Energy & The Pendulum Units • Newton = ?

  22. Conservation of Energy & The Pendulum Units • Newton = kg-m/sec^2 • Energy • Newton-m • Kg-m^2/sec^2

  23. Conservation of Energy Energy is conserved • PE + KE = constant For springs, • PE = ½ kx2 For objects in motion, • KE = ½ mv2

  24. Conservation of Energy & The Pendulum • http://zonalandeducation.com/mstm/physics/mechanics/energy/springPotentialEnergy/springPotentialEnergy.html

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