Springs And pendula, and energy

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

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 (F a x). Hooke’s Law: Felastic = -kx Where: k = spring constant = stiffness of spring (N/m) x = displacement

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.

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 W = ½ kx2 = D PE + D KE 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

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

Simple Harmonic Motion & Springs 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.

Restoring Forces and 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

Simple Harmonic Motion & Springs

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.

10.3 Energy and Simple Harmonic Motion Example 3 Changing the Mass of a Simple Harmonic Oscilator 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?

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

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