3.6.1 Periodic Motion
3.6.1.1 Circular Motion
● Angular frequency is the rate of change of angle (radians per second)
θ
ω= t
= 2πf
● Angular velocity = how fast the object is rotating
|v| = ωr (where r = radius)
Centripetal Acceleration
● Centripetal force is the force which acts on the body towards the center of
the circle of its motion
● Centripetal acceleration is the acceleration of an object moving in circular
motion
● Called centripetal acceleration because the acceleration is always towards
the center of the circle of motion
v2
ac = r
= ω2
mv 2
F Centripetal = r
= mω 2 r
3.6.1.2 Simple Harmonic Motion (SHM)
● A body will undergo simple harmonic motion when the force that tries to
restore the object to its rest position is proportional to the displacement of
the object
● F = − kx
● This motion can be shown by a sine wave (or cosine wave)
x = Acos(ωt)
v = − ω Asin(ωt)
a = − ω 2 Acos(ωt) = − ω 2 x
M ax value of cos(ωt) and sin(ωt) = 1
⇒ M ax speed = ωA
⇒ M ax acceleration = ω 2 A
,3.6.1.3 Simple Harmonic Systems
● A spring is an example of a simple harmonic system
● In a spring system:
√
m
T = 2π k (where k is the spring constant)
● For a pendulum:
√
l
T = 2π g (where l is the length of the pendulum)
● Pendulum equation only works for small angles (see below for reason)
Derivation for Time Period of a Pendulum
F = − kx
− mg sinθ = − k x
mg sinθ
k = x
Arc length s = θr = θl (radius = length of pendulum)
x ≈ s ⇒ x = θl
mg sinθ
k = θl
S M ALL AN GLE AP P ROXIM AT ION S − θ ≈ sinθ, so cancel
mg
k = l
√
m
T = 2π k
m
T = 2π
√ mg
l
√
ml
T = 2π mg
√
l
T = 2π g
Aside
, F = kx
ma = kx
m (ω 2 Acos(ωt)) = k(Acos(ωt))
k
m = ω2
SHM and Energy
E T otal = E P otential + E Kinetic
● As the angle of the pendulum increases, kinetic energy decreases and
potential energy decreases
● As the angle decreases, kinetic energy increases and kinetic energy
decreases
● At the origin position ( x = 0 ), kinetic energy (and so velocity) is at a
maximum (no potential energy)
1 2
ET = 2
kx + 12 mv 2
ET = 1
2
k(A2 cos2 (ωt)) + 12 ( ωk2 )(A2 ω 2 sin2 (ωt))
ET = 1
2
kA2 (cos2 (ωt) + sin2 (ωt))
ET = 1
2
kA2
● From this we can find a formula for v :
1
2
kA2 = 12 kx2 + 1
2
mv 2
mv 2 = kA2 − kx2
( ωk2 )v 2 = k(A2 − x2 )
v 2 = ω 2 (A2 − x2 )
v = ± ω √A2 − x2
v max at x = 0
⇒ v max = ± ω A
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller JodbyBerundi. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £2.99. You're not tied to anything after your purchase.
