Circular Motion

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Circular Motion
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Imagine a hammer (athletics variety) being spun
in a horizontal circle

• At a constant speed

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Birds-Eye View
?
v
r
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v r?
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Side View
?
T
mg
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We know that the hammer is accelerating..

• Because the hammer is constantly changing
  direction (although the speed is constant)

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So from Newtons First and Second Laws, there
must be a resultant force

• Equal to
• mass x acceleration

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For circular motion..

• Acceleration v2
• r
• or r?2 (using v r? )

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So the resultant force ..

• mv2
• r
• or mr?2 (using v r? )

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Which direction do the resultant force and
acceleration act in?

• Towards the centre of the described circle

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So if we look at our original diagram.
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If the circle has a radius ,r.
?
T
mg
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We can find the resultant force by resolving in
the plane of the circle.

• The only force acting in the horizontal plane is
  the tension
• So by resolving T mr?2

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Very important point!

• The circular force is not an additional force
  it is the resultant of the forces present.

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Typical exam style question

• Ball hangs from a light piece of inextensible
  string and describes a horizontal circle of
  radius,r and makes an angle ? with the vertical .
• If the mass of the ball is m kg
• calculate the tension, T in the string
• calculate the angular velocity, ? in terms of g,
  r and ?.

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Diagram
?
T
mg
r
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To find the tension.

• Resolve vertically
• Ball is not moving up or down so vertical
  components must be equal

Tcos? mg so T mg cos?
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To find the angular velocity, ?

• Resolve horizontally
• Circular motion so we know that there is a
  resultant force towards the centre

Tsin? mr?2 ? Tsin? mr
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But

• T mg
• cos?

so ? Tsin? mr Becomes ?
gtan? r
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Easy?