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Chapter

# Circular Motion.docx

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Queen's University

Physics

PHYS 106

Alastair B Mc Lean

Fall

Description

Uniform circular motion
- motion at a constant SPEED around a circle
- in other words, the velocity vector is always equal in magnitude, and always tangent to the circle
Period (T): the time it takes for an object to make one revolution, in seconds. Eg. for 250 rps, T=1/250
Thus for uniform circular motion: |v|= 2πr/T
(circumference of a circle over time)
Angular position
- we can describe the position of a particle in circular motion by the radius of the circle and the angle it
makes counterclockwise from the x-axis
- relationship between arc length, angle, and radius:
s= θr
where r is in radians.
180* = π
Use this to convert between degrees and radians.
Angular displacement Δθ and angular velocity ω
- the change in a particle's position in circular motion can be measured by finding the change in its angle,
Δθ: the angular displacement
- the average change in a particle's angular displacement over time, is Δθ/Δt
Average angular velocity: Δθ/Δt
Taking the limit as t-> 0 gives the:
Instantaneous angular velocity: ω= dθ/dt, in rad/s
- this is the rate at which a particle's ANGULAR position is changing as it moves around the circle
- uniform circular motion can also now be defined as motion that has constant angular velosity
- counterclockwise = positive direction
The area (ω Δt) under the ω vs t curve, between t1 and t2 is θ 12
Thus θ 12ω Δt
θ2= θ1+ ω Δt
Notice that all these equations parallel the definitions of displacement and velocity in non-accelerating
linear motion:
D=vt
x12 v Δt
x2= x + vΔt
1
When the angular acceleration is constant/ 0, you can just use the constant acceleration kinematic
equations: ω= x rpm; Conversions to rad/s
1) Multiply by 1 min/60s
2) multiply by 2pi/1 rev
Finding the position of a particle after multiple revolutions
1) Use the equation θ =2θ + 1 Δt to solve for θ 2.
2) Divide θ b2 2π to find the number of revolutions.
3) θ2= number of complete revolutions * 2 π + fraction of incomplete revolution * 2π
4) But from an observer's point of view, the particle's position is at fraction of incomplete revolution * 2π.
Velocity ⃗⃗ in Uniform circular motion
- velocity's magnitude is constant, but its direction is always changing such that it remains tangent to the
circle
- in other words, the velocity vector only has a tangential component, and
the tangential velocity = ωr in m/s
⃗⃗= (vr, [email protected]) = (0, ωr)
Acceleration ⃗⃗in uniform circular motion
- although objects in circular motion move with a
constant speed, the velocity changes in direction
(constantly), and thus the object has an acceleration
- to keep the object moving in a circle, a force must
be applied to the object
- if no force was acting on object (red): it would fly
out of the circular orbit (see purple path)
- green arrows show the direction of this force that
keeps it

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