1. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 INTEGRATION APPLICATIONS 3 PROGRAMME 21

2. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Moments of inertia Second moments of area Centre of pressure

3. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Moments of inertia Second moments of area Centre of pressure

4. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Moments of inertia Introduction Radius of gyration Parallel axes theorem Perpendicular axes theorem (for thin plates) Useful standard results

5. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Moments of inertia Introduction 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 ω then: and so:

6. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 The total kinetic energy of n particles, each of mass m i moving in a circle about a fixed axis perpendicular to the circle of radius r i and each with the same angular velocity ω is given as: Where: the moment of inertia (or second moment of mass) of the total mass Moments of inertia Introduction

7. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 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: Moments of inertia Radius of gyration

8. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 To find the moment of inertia and radius of gyration of a uniform thin rod of length a and linear mass density ρ 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 δx is ρδx so that the moment of inertia of the element is: Hence: and in the limit as δx → 0: Moments of inertia Radius of gyration

9. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Since: then: Moments of inertia Radius of gyration

10. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 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. Moments of inertia Parallel axes theorem

11. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Let δm be an element of mass at P. Then: so that: Moments of inertia Perpendicular axes theorem (for thin plates)

12. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Rectangular plate Circular disc Moments of inertia Useful standard results

13. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Parallel axes theorem Perpendicular axes theorem Moments of inertia Useful standard results

14. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Moments of inertia Second moments of area Centre of pressure

15. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Moments of inertia Second moments of area Centre of pressure

16. STROUD Worked examples and exercises are in the text Programme 21: 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 ’.

17. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Second moments of area Moments of inertia Second moments of area Rectangular plate Rectangle

18. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Second moments of area Moments of inertia Second moments of area Circular plate Circle

19. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Second moments of area Moments of inertia Second moments of area Parallel axes theorem Perpendicular axes theorem

20. STROUD Worked examples and exercises are in the text Programme 21: 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 I 1, I 2, I 3,..., then the second moment of the composite figure I about the same axis is simply the sum

21. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Moments of inertia Second moments of area Centre of pressure

22. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Moments of inertia Second moments of area Centre of pressure

23. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Centre of pressure Pressure at a point P, depth z below the surface of a liquid Total thrust on a vertical plate immersed in liquid Depth of the centre of pressure

24. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Centre 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.

25. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Centre 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.δz Total thrust on the plate: In the limit as δz → 0

26. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Centre of pressure Total thrust on a vertical plate immersed in liquid Total thrust on the plate:

27. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Centre of pressure Depth of the 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:

28. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 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 Centre of pressure Depth of the centre of pressure

29. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 Sum of the moments of the thrust about the surface is then: In the limit as δz → 0 this becomes Centre of pressure Depth of the centre of pressure

30. STROUD Worked examples and exercises are in the text Programme 21: Integration applications 3 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: Centre of pressure Depth of the centre of pressure

31. STROUD Worked examples and exercises are in the text Programme 21: 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