Motion in a circle

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flipperworks.com 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 = 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 = 2 and a = to solve

Transcript of Motion in a circle

Page 1: Motion in a circle

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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.

Page 2: Motion in a circle

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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

Page 3: Motion in a circle

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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

Page 4: Motion in a circle

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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

Page 5: Motion in a circle

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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)

Page 6: Motion in a circle

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

Page 7: Motion in a circle

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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)

Page 8: Motion in a circle

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Page 9: Motion in a circle

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