SME
Dynamics · Problem type
A sliding ladder by its instantaneous centre
Use this for “A 5 m ladder slides with its foot moving out at 2 m/s when it makes 60° with the floor. Find ω and the top’s speed.”
Diagram
The
IC
is
where
the
normals
to
the
two
paths
meet:
above
the
foot,
level
with
the
top.
ω
=
vA
L sin θ
=
0.462
rad/s.
vB
=
vA
tan θ
=
1.15
m/s,
down
the
wall.
This is an example. Type your own values to start a new problem.
ω (rad/s)
=
vA (m/s)
÷
(
L (m)
sin
θ
°)
vB (m/s)
=
vA (m/s)
÷
tan
θ
°
rA
Foot to IC
worked out (not typed) · m
rB
Top to IC
worked out (not typed) · m
Formulas
ω
=
vA
L
× sin(
θ
)
0.4619
=
2
5 × sin(60°)
vB
=
vA
tan(
θ
)
1.1547
=
2
tan(60°)
rA
=
L
×
sin(
θ
)
4.3301 = 5 × sin(60°)
rB
=
L
×
cos(
θ
)
2.5 = 5 × cos(60°)
Assumptions
•
The foot slides along the floor and the top down the wall, both staying in contact.
•
The IC is where the normals to those paths meet: L sin θ above the foot, L cos θ from the top.
•
θ is the ladder’s angle with the floor; the top moves down while the foot moves out.
Step-by-step
WE KNOW
L = 5 m, θ = 60°, vA = 2 m/s
FIND
angular speed (ω), top speed (down) (vB), foot to IC (rA), top to IC (rB)
Step 1 · Find angular speed (ω)
Use ω = vA ÷ (L × sin(θ))
The
foot
is
L
sin
θ
from
the
IC
and
moves
at
ω
times
that
distance:
ω
=
v
r
.
ω
=
2
5 × sin(60°)
ω
=
2
5 × 0.866
ω
=
2
4.33
ω
=
0.4619
rad/s
Step 2 · Find top speed (down): vB
Use vB = vA ÷ tan(θ)
Both
ends
turn
about
the
IC
at
the
same
ω
,
so
their
speeds
go
as
their
distances:
vB
vA
=
L
cos
θ
L
sin
θ
.
vB
=
2
tan(60°)
vB
=
2
1.7321
vB = 1.1547 m/s
Step 3 · Find foot to IC (rA)
Use rA = L × sin(θ)
The
IC
is
level
with
the
top,
so
it
sits
the
ladder’s
height
L
sin
θ
above
the
foot.
rA = 5 × sin(60°)
rA = 5 × 0.866
rA = 4.3301 m
Step 4 · Find top to IC (rB)
Use rB = L × cos(θ)
The
IC
is
straight
above
the
foot,
so
it
sits
the
foot’s
distance
from
the
wall,
L
cos
θ
,
from
the
top.
rB = 5 × cos(60°)
rB = 5 × 0.5
rB = 2.5 m
Check
Put the numbers back in. Both sides should match.
0.4619
=
2
5 × sin(60°)
✓
1.1547
=
2
tan(60°)
✓
4.3301 = 5 × sin(60°) ✓
2.5 = 5 × cos(60°) ✓
Topic check
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