Springs And pendula, and energy

Spring Constants SpringkUnits Small Spring Long Spring Medium spring 2 in series 2 in parallel 3 in series 3 in parallel 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 (F  x). Hooke’s Law: F elastic = -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. Add Hooke’s law problems Add graph, show work on graph as area under triangle

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 =  PE +  KE

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

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

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: 1. This formula is true for only small angles of θ. 2. 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 Pe g = F g * 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? Pe g = F g * 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 kx 2 For objects in motion, KE = ½ mv 2

Conservation of Energy & The Pendulum Conservation of Mechanical Energy PE i + KE i = PE f + KE f mgΔh = ½ mv 2 gΔh = ½ v 2 If you solve for v: v = √ 2gΔh v = √ 2(9.81 m/s 2 )(0.45 m) v = 2.97 m/s

Conservation of Energy & The Pendulum mechanics/energy/springPotentialEnergy/spri ngPotentialEnergy.html mechanics/energy/springPotentialEnergy/spri ngPotentialEnergy.html