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## Springs

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**Springs**And pendula, and energy**Spring Constants**Do these results make sense based on your sense of spring “stiffness”?**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**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.**Hooke’s Law – Energy**• F = kx • W = Fd • W = (average F)d • W = [F(final) – F(initial)]/2*d • W = [kx - 0 ]/2*x • W = ½ kx^2 = D PE + D KE**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**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**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.**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**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.**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**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**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**Conservation of Energy**• What is the equilibrium position of the ball? • How far will it fall before being pulled • Back up by the spring?**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**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?**Conservation of Energy & The Pendulum**Units • Newton = ?**Conservation of Energy & The Pendulum**Units • Newton = kg-m/sec^2 • Energy • Newton-m • Kg-m^2/sec^2**Conservation of Energy**Energy is conserved • PE + KE = constant For springs, • PE = ½ kx2 For objects in motion, • KE = ½ mv2**Conservation of Energy & The Pendulum**• Conservation of Mechanical Energy • PEi + KEi = PEf + KEf • mgΔh = ½ mv2 • gΔh = ½ v2 • If you solve for v: • v = √ 2gΔh • v = √ 2(9.81 m/s2)(0.45 m) • v = 2.97 m/s**Conservation of Energy & The Pendulum**• http://zonalandeducation.com/mstm/physics/mechanics/energy/springPotentialEnergy/springPotentialEnergy.html