Horizontal mass-spring system - position, velocity, and acceleration curves.

1. Physics
101
learning
object
Simple
harmonic
motion:
position,
velocity,
and
acceleration
A
horizontal
mass
spring
system
is
compressed
to
position
A,
and
let
go.
Q1:
Consider
the
position
versus
time
graph
given
by
𝑥 𝑡 = 3cos (𝑥 + 𝜋)
below.
Assuming
that
the
object
moves
on
a
frictionless
surface
in
vacuum;
sketch
velocity
and
acceleration
curves
on
the
set
of
axes
below.
Q2:
In
which
position
(A,
B,
C,
D,
Equilibrium)
does
maximum
positive
velocity
occur?
In
which
position
(A,
B,
C,
D,
Equilibrium)
does
maximum
acceleration
occur?
A
B
C
D
A
http://media.cheggcdn.com/media/89a/89a99d34-­‐1f57-­‐433c-­‐83db-­‐6bed623b0d45/phpXI1h0v.png

2. Physics
101
learning
object
:
Explanations
Simple
harmonic
motion:
position,
velocity,
and
acceleration
A
horizontal
mass
spring
system
is
compressed
to
position
A,
and
let
go.
Q1:
Consider
the
position
versus
time
graph
given
by
𝑥 𝑡 = 3cos (𝑥 + 𝜋)
below.
Assuming
that
the
object
moves
on
a
frictionless
surface
in
vacuum;
sketch
velocity
and
acceleration
curves
on
the
set
of
axes
below.
• Recognising
that
velocity
(green)
is
the
derivative
of
position:
𝑣 𝑡 = −3 sin 𝑥 + 𝜋 ,
and
that
the
velocity
curve
is
0.5π
out
of
phase
• Recognising
that
acceleration
(black)
is
the
second
derivative
of
position:
𝑎 𝑡 = − 3cos 𝑥 + 𝜋 ,
and
that
the
acceleration
curve
is
π
out
of
phase
Q2:
At
which
position
(A,
B,
C,
D,
Equilibrium)
is
the
velocity
zero?
At
which
position
(A,
B,
C,
D,
Equilibrium)
is
the
acceleration
zero?
• Zero
velocity
occurs
at
positions
A
and
D,
because
the
object
changes
direction,
thus
the
velocity
changes
sign
and
is
momentarily
zero.
• Zero
acceleration
occurs
at
equilibrium
position.
Due
to
the
restoring
force
of
the
spring
that
always
pulls
the
object
in
the
opposite
direction
that
it
is
travelling,
acceleration
is
at
all
times
in
opposite
direction
as
displacement.
At
equilibrium
position,
the
restoring
force
is
zero,
hence
acceleration
too
is
zero.
A
B
C
D
A
http://media.cheggcdn.com/media/89a/89a99d34-­‐1f57-­‐433c-­‐83db-­‐6bed623b0d45/phpXI1h0v.png