1. Learning
Object:
Vertical
Oscillation
in
simple
harmonic
motion
Question:
A
metal
ruler
is
placed
on
a
desk
and
its
bent
undergoing
simple
harmonic
motion
with
a
frequency
of
6
cycles
per
second.
What
is
the
maximum
amplitude
with
which
the
end
of
the
board
can
fluctuate
in
order
to
keep
a
skittle
on
the
edge
to
not
fall
off?
A) 0.05m
B) 0.03m
C) 0.01m
D) 0.04m
Solution:
X(t)=
A*
cos(wt)
Frequency=
4
cycles
per
second-­‐-­‐-­‐-­‐-­‐
w=
2pi*f=
6pi
We
Take
out
simple
harmonic
equation
and
plug
in
W
X(t)=
A*cos((6pi)*t)
We
now
need
to
find
acceleration
in
order
to
see
the
capacity
of
the
skittle
staying
on
the
ruler.
Acceleration
is
the
second
time
derivative
of
the
function
above
Dx^2/Dt^2=
-­‐A*6pi*6pi*cos(6pi*t)=-­‐(6pi)^2*A*cos(6pi*t)
It
will
reach
a
maximum
force
of
(6pi)^2*A
If
the
skittle
is
to
stay
on
the
ruler,
the
gravitational
force
must
be
equal
to
the
force
exerted
upwards
by
the
ruler
F=ma
F=
(6pi)^2A*m
Now
just
solve
for
the
missing
variable
Amplitude
A=g/[(6pi)^2]
A=
9.8/[(6pi)^2]=
0.03m
The
maximum
amplitude
that
a
ruler
can
move
is
0.03m