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Circular Motion

Understand the forces involved in circular motion, specifically the tension in a horizontal circle and the calculation of angular velocity. Learn how to solve related exam-style questions.

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Circular Motion

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  1. Circular Motion

  2. Imagine a hammer (athletics variety) being spun in a horizontal circle At a constant speed

  3. Birds-Eye View ω v r

  4. v = rω

  5. Side View ω T mg

  6. We know that the hammer is accelerating….. Because the hammer is constantly changing direction (although the speed is constant)

  7. So from Newton’s First and Second Laws, there must be a resultant force Equal to mass x acceleration

  8. For circular motion….. Acceleration = v2 r or rω2 (using v = rω )

  9. So the resultant force ….. = mv2 r or mrω2 (using v = rω )

  10. Which direction do the resultant force and acceleration act in? Towards the centre of the described circle

  11. So if we look at our original diagram…….

  12. If the circle has a radius ,r…. ω T mg

  13. 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 = mrω2

  14. Very important point! The ‘circular force’ is not an additional force – it is the resultant of the forces present.

  15. 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 θ with the vertical . If the mass of the ball is m kg • calculate the tension, T in the string • calculate the angular velocity, ω in terms of g, r and θ.

  16. Diagram θ T mg r

  17. To find the tension…. Resolve vertically Ball is not moving up or down so vertical components must be equal Tcosθ = mg so T = mg cosθ

  18. To find the angular velocity, ω… Resolve horizontally Circular motion so we know that there is a resultant force towards the centre Tsinθ = mrω2 ω =Tsinθ mr

  19. But…… T = mg cosθ so ω = Tsinθ mr Becomes ω = gtanθ r

  20. Easy?

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