ussc3002 oscillations and waves lecture 1 one dimensional systems wayne m. lawton department of...
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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 [email protected]://www.math.nus/~matwml
Tel (65) 6874-2749
1
NEWTON’S SECOND LAW
The net force on a body is equal to the product of the body’s 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?
s/m6v
amF
2
STATICS
Why is this object static (not moving) ?
mg
What are the forces acting on this object?What is the net force acting on this object? 3
VECTOR ALGEBRA FOR STATICS
The tension forces are
mg
0F
g
θsina
θcosaF
l
sinb
cosbF
r
The gravity force is
4
TUTORIAL 1
1. Analyse the forces on an object that slides down a frictionless inclined plane. What is the net force?
θh
Compute the time that it takes for an object with initial speed zero to slide down the inclined plane.
5
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 f
i
x
xdxxFW )(
Newton’s 2nd Law
dx
dvv
dt
dx
dx
dv
dt
dvxa )(Chain Rule
)()( xmaxF
)()(if
f
i
xx TTvdvmWx
x)(2
21)( xmvxT Kinetic Energy
6
POTENTIAL ENERGY
Definition
and in that case we can also compute the work as
)()()( fi
x
xxVxVdxxFW
f
i
V is a potential energy function if
VTE
dx
xdVxF
)()(
so the total energy is constant since
)()()()()()( iiifff xExVxTxVxTxE 7
VIBRATIONS IN A SPRINGFor an object attached to a spring that moves horizontally, the total energy is
22
21
21 xmkxE
)t(cosa)t(x xx 0
kE2a is conserved, thereforewhere
mkR
2T
is the angular frequency
is the phase, and
is the period.
is the amplitude
(where )/ dtdxx
8
NONLINEAR VIBRATIONS
Since the energy for a pendulum
θcosθ22
2
LmgE mL θ L
is the constant
we can compute
2
)cos(2
dt
dθ
mL
LmgE
θ from the nonlinear ODE
LmgE mL )0(θ22
2
9
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.
4. Derive an approximation for E for pendulum if
0θmax
and use it to derive a linear approximation for the equation of motion for a pendulum.
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 ODE’s.
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