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PROGRAMME 20

PROGRAMME 20. INTEGRATION APPLICATIONS 3. Programme 20: Integration applications 3. Moments of inertia Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results Second moments of area Centres of pressure Depth of centre of pressure.

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PROGRAMME 20

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  1. PROGRAMME 20 INTEGRATION APPLICATIONS 3

  2. Programme 20: Integration applications 3 Moments of inertia Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results Second moments of area Centres of pressure Depth of centre of pressure

  3. Programme 20: Integration applications 3 Moments of inertia Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results Second moments of area Centres of pressure Depth of centre of pressure

  4. Programme 20: Integration applications 3 Moments of inertia The kinetic energy of a single particle of mass m moving with velocity v is given as: If the body is rotating in a circle of radius r with an angular velocity  then: and so:

  5. Programme 20: Integration applications 3 Moments of inertia The total kinetic energy of n particles, each of mass mi moving in a circle about a fixed axis perpendicular to the circle of radius ri and each with the same angular velocity  is given as: Where: the moment of inertia (or second moment of mass) of the total mass

  6. Programme 20: Integration applications 3 Moments of inertia Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results Second moments of area Centres of pressure Depth of centre of pressure

  7. Programme 20: Integration applications 3 Radius of gyration If the n particles have a total mass M where M is taken to be located at a distance k from the fixed point such that the K.E of the total mass M is the same as the total K.E of the distributed particles then: So that:

  8. Programme 20: Integration applications 3 Radius of gyration To find the moment of inertia and radius of gyration of a uniform thin rod of length a and linear mass density  about an axis through one end and perpendicular to the rod it is noted that the mass of an element of the rod of length x is x so that the moment of inertia of the element is: Hence: and in the limit as x→ 0:

  9. Programme 20: Integration applications 3 Radius of gyration Since: then:

  10. Programme 20: Integration applications 3 Moments of inertia Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results Second moments of area Centres of pressure Depth of centre of pressure

  11. Programme 20: Integration applications 3 Parallel axes theorem If the moment of inertia is known about an axis through the centre of gravity of an object then it is a simple matter to find the moment of inertia about any other axis parallel to it.

  12. Programme 20: Integration applications 3 Moments of inertia Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results Second moments of area Centres of pressure Depth of centre of pressure

  13. Programme 20: Integration applications 3 Perpendicular axes theorem (for thin plates) Let m be an element of mass at P. Then: so that:

  14. Programme 20: Integration applications 3 Moments of inertia Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results Second moments of area Centres of pressure Depth of centre of pressure

  15. Programme 20: Integration applications 3 Useful standard results Rectangular plate Circular disc

  16. Programme 20: Integration applications 3 Useful standard results Parallel axes theorem Perpendicular axes theorem

  17. Programme 20: Integration applications 3 Moments of inertia Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results Second moments of area Centres of pressure Depth of centre of pressure

  18. Programme 20: Integration applications 3 Second moments of area The second moment of area has nothing to do with the kinetic energy of rotation but the mathematics involved is very much akin to that for moments of inertia: indeed, the same symbol I is used for both. In the calculations, the ‘mass’ is replaced by ‘area’.

  19. Programme 20: Integration applications 3 Second moments of area Moments of inertia Second moments of area Rectangular plateRectangle

  20. Programme 20: Integration applications 3 Second moments of area Moments of inertia Second moments of area Circular plateCircle

  21. Programme 20: Integration applications 3 Second moments of area Moments of inertia Second moments of area Parallel axes theorem Perpendicular axes theorem

  22. Programme 20: Integration applications 3 Second moments of area Composite figures If a figure is made up of a number of figures whose individual second moments about a given axis are I1, I2, I3, . . . , then the second moment of the composite figure I about the same axis is simply the sum

  23. Programme 20: Integration applications 3 Moments of inertia Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results Second moments of area Centres of pressure Depth of centre of pressure

  24. Programme 20: Integration applications 3 Centres of pressure Pressure at a point P, depth z below the surface of a liquid Total thrust on a vertical plate immersed in liquid

  25. Programme 20: Integration applications 3 Centres of pressure Pressure at a point P, depth z below the surface of a liquid For a perfect liquid the pressure p at P (the thrust on unit area) is due to the weight of the column of liquid of height z above it (ignoring atmospheric pressure). Pressure at P is: where w is the weight of unit volume of the liquid.

  26. Programme 20: Integration applications 3 Centres of pressure Total thrust on a vertical plate immersed in liquid Pressure at P is wz Thrust on strip PQ wz× (area of strip) wz.a.z Total thrust on the plate: In the limit as z → 0

  27. Programme 20: Integration applications 3 Centres of pressure Total thrust on a vertical plate immersed in liquid Total thrust on the plate:

  28. Programme 20: Integration applications 3 Moments of inertia Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results Second moments of area Centres of pressure Depth of centre of pressure

  29. Programme 20: Integration applications 3 Depth of centre of pressure The pressure on an immersed plate increases with depth. The total thrust T can be considered as a resultant force acting at a point Z called the centre of pressure. The depth of the centre of pressure is denoted by:

  30. Programme 20: Integration applications 3 Depth of centre of pressure To locate the centre of pressure take moments of forces about the axis where the plane of the plate cuts the surface of the liquid. Thrust on strip PQ Moment of the thrust about the surface is then

  31. Programme 20: Integration applications 3 Depth of centre of pressure Sum of the moments of the thrust about the surface is then: In the limit as z→ 0 this becomes

  32. Programme 20: Integration applications 3 Depth of centre of pressure The sum of the moments of the thrust about the surface is equal to the total moment of the total thrust about the centre of pressure. That is: and so: Hence:

  33. Programme 20: Integration applications 3 Learning outcomes • Determine moments of inertia • Determine the radius of gyration • Use the parallel axes theorem • Use the perpendicular axes theorem for thin plates • Determine moments of inertia using standard results • Determine second moments of area • Determine centres of pressure

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