Springs

And pendula, and energy

Spring ConstantsSpring k Units

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: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.

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 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

Simple Harmonic Motion & Springs

The Pendulum

Like a spring, pendula go through simple harmonic motion as follows.

T = 2π√l/gWhere: 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

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

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