USSC3002 Oscillations and Waves Lecture 1 One Dimensional Systems

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USSC3002 Oscillations and Waves Lecture 1 One
Dimensional Systems

• Wayne M. Lawton
• Department of Mathematics
• National University of Singapore
• 2 Science Drive 2
• Singapore 117543

Email matwml_at_nus.edu.sg http//www.math.nus/matwm
l Tel [phone]
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NEWTONS SECOND LAW
The net force on a body is equal to the product
of the bodys mass and the acceleration of the
body.
Question what constant horizontal force must be
applied to make the object below (sliding on a
frictionless surface) stop in 2 seconds?
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STATICS
Why is this object static (not moving) ?
What are the forces acting on this object? What
is the net force acting on this object?
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VECTOR ALGEBRA FOR STATICS
The tension forces are
The gravity force is

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TUTORIAL 1
1. Analyse the forces on an object that slides
down a frictionless inclined plane. What is the
net force?
Compute the time that it takes for an object with
initial speed zero to slide down the inclined
plane.
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WORK-KINETIC ENERGY THEOREM
Consider a net force that is applied to an object
having mass m that is moving along the x-axis
The work done is
Newtons 2nd Law
Chain Rule
Kinetic Energy
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POTENTIAL ENERGY
Definition
is a potential energy function if
and in that case we can also compute the work as
so the total energy
is constant since
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VIBRATIONS IN A SPRING
For an object attached to a spring that moves
horizontally, the total energy is
(where
is conserved, therefore
where
is the amplitude
is the angular frequency
is the phase, and
is the period.
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NONLINEAR VIBRATIONS
Since the energy for a pendulum
L
is the constant
we can compute
from the nonlinear ODE
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TUTORIAL 1
2. Show that the solution x(t) of the spring
problem satisfies a second order linear ODE and
derive this ODE directly from the equation E
constant.
3. Show directly that the set of solutions of the
ODE in problem 2 forms a vector space (is closed
under multiplication by real numbers and under
addition). This is the case for all linear ODEs.
4. Derive an approximation for E for pendulum if
and use it to derive a linear approximation for
the equation of motion for a pendulum.
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