SME
Grade 11 Science · Physics
Simple harmonic motion: springs and pendulums
Use this for “A 0.5 kg mass bounces on a 200 N/m spring with a 0.1 m amplitude. What are its period, frequency and top speed?”
Diagram
T
=
2π
√
m/k
=
2π
×
√
0.5/200
T = 0.3142 s
vₘₐₓ = Aω = 0.1 × 20 = 2 m/s
It stops at ±A and moves fastest through x = 0; a bigger A leaves T the same.
This is an example. Type your own values to start a new problem.
T (s)
=
2π
m (kg)
k (N/m)
f
Frequency
calculated · Hz
ω
Angular frequency
calculated · rad/s
A
Amplitude
example
vₘₐₓ
Top speed
calculated
E
Energy
calculated · J
Formulas
T
=
2π
×
√
m
k
0.3142
=
2π
×
√
0.5/200
ω
=
√
k
m
20
=
√
200/0.5
f
=
1
T
3.18 = 1/0.3142
ω
=
2π/
T
20 = 2π/0.3142
vₘₐₓ
=
A
×
ω
2 = 0.1 × 20
E
=
1
2
×
k
×
A
2
1
=
1
2
×
200
×
0.1
2
Assumptions
•
The spring obeys Hooke’s law (F = −kx) and nothing rubs, so the swing never dies down.
•
The period depends only on m and k: pulling it farther (a bigger A) doesn’t change T.
•
It moves fastest through the middle and stops for an instant at each end.
Step-by-step
WE KNOW
m = 0.5 kg, k = 200 N/m, A = 0.1 m
FIND
period (T), angular frequency (ω), frequency (f), top speed (vₘₐₓ), energy (E)
Step 1 · Find period (T)
Use T = 2π × √(m/k)
A heavier mass swings more slowly; a stiffer spring, faster.
T
=
2π
×
√
0.5/200
√
0.0025
=
0.05
T
=
2π
×
0.05
=
0.1π
T
=
0.3142
s
Step 2 · Find angular frequency (ω)
Use ω = √(k/m)
A
stiffer
spring
or
a
lighter
mass
swings
faster:
the
square
root
of
k
over
m
.
ω
=
√
200/0.5
ω
=
√
400
ω
=
20
rad/s
Step 3 · Find frequency (f)
Use f = 1/T
The frequency is how many swings fit in one second.
f
=
1/0.3142
f
=
3.18
Hz
Step 4 · Find top speed (vₘₐₓ)
Use vₘₐₓ = A × ω
Through
the
middle
it
moves
fastest:
the
amplitude
times
ω
.
vₘₐₓ = 0.1 × 20
vₘₐₓ = 2 m/s
Step 5 · Find energy (E)
Use E = ½ × k × A²
At
each
end
it
is
all
spring
energy,
stretched
by
A
.
E
=
1
2
×
200
×
0.1
2
E
=
1
2
×
200
×
0.01
E
=
100
×
0.01
E
=
1
J
Check
Put the numbers back in. Both sides should match.
0.3142
≈
2π
×
√
0.5/200
✓
20
=
√
200/0.5
✓
3.18 ≈ 1/0.3142 ✓
20 ≈ 2π/0.3142 ✓
2 = 0.1 × 20 ✓
1
=
1
2
×
200
×
0.1
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.