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Assessment Objectives (for Motion in a Circle):
Students should be able to:a) express angular displacement in radians ( θ = ).
b) understand and use the concept of angular velocity, ω to solve problems.
c) recall and use v = rω to solve problems.d) describe qualitatively motion in a curved path due to a
perpendicular force, and understand the centripetal acceleration in the case of uniform motion in a circle.
e) recall and use centripetal acceleration a = rω2 and a = to solve problems.
f) recall and use centripetal force F = mrω2 and F = to solve problems.
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θ = = r
sarc ofradius
length arc
Angular displacement is defined:
Angular Displacement
To convert in degrees to radians: (rad) =
Note: rad is physically dimensionless as it is the ratio of two lengths.
2
360o
o
θr
s
and θ = 1 radian (rad) when s = r
For one complete revolution,
θ =
= = 2π radarc of radius
circle of cir.
rr2
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ω
Angular VelocityAngular velocity ω is defined as the rate of change of angular displacement or the change in angular displacement per unit time.
Δθ/Δt = ω
r
vf at t2 vi at t1
td
d i.e. ω = unit of ω : rad s-1
“Omega”
Note: Angular velocity, ω is a vector quantity.direction of
rotation
direction of rotation
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Angular VelocityAngular velocity ω is defined as the rate of change of angular displacement or the change in angular displacement per unit time.
ω = td
d
Δθ/Δt = ω
r
vf at t2 vi at t1
td
d i.e. ω = unit of ω : rad s-1
“Omega”
Consider an object moving with speed v in a circular path of radius r:
Note: Angular velocity, ω is a vector quantity.
v = rω= 1
r
v
dt
ds
rr
s
dt
d
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For uniform circular motion, since v has the same magnitude throughout the motion, angular velocity is a constant.
T is known as the period. Do you know how T and f is related?
T
2 ω
If T is the time taken to complete one revolution then
T f
1 unit of f : Hertz (Hz)
flipperworks.comCentripetal Acceleration In a uniform circular motion, the velocity is always changing, the motion is accelerated and by Newton’s second law, a resultant force must be acting on it.
Δv
- vi
vf
Δv = vf – vi
= vf + (- vi)
a = = rω2
a = = = since = for Δt → 0
viA B
C
vf
where a is the centripetal acceleration
Δv
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The force causing the circular motion is known as the centripetal force and is given by
F = = mrω2 r
mv2
Points to note:• centripetal force should not be drawn as an additional
force in force diagrams• if there is no centripetal force an object in circular
motion would fly off in a direction tangent to the circular path
• since the centripetal force acts at right angles to the motion, the centripetal force does no work (centripetal force does not change the speed of the object)
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