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Circular Motion.docx

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Queen's University
PHYS 106
Alastair B Mc Lean

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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