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Centre of mass of a uniform plane lamina
- To find the distance x' of the centre of mass of a lamina from the y-axis we have:
having both integrals between the x-values as x=a and x=b, where y = f(x).
- To find the distance y' of the centre of mass of the lamina from the x-axis we have:
Having both integrals between the x-values as x=a and x=b, where y = f(x)
Centre of mass of a uniform solid body
- The centre of mass of a solid body is the point at which the weight acts.
- The weight of a uniform solid body is evenly distributed throughout its volume. So we define a constant ρ as the mass per unit volume.
- The centre of mass of a uniform solid body must lie on any axis of symmetry.
- The centre of mass of a uniform solid body must lie on any plane of symmetry.
Use of symmetry
- Uniform solid sphere - The sphere has an infinite number of planes of symmetry, so the centre of mass of the
sphere is directly in the centre.
- Uniform solid right circular cylinder - The line through the plane face perpendicular to this face is an axis of symmetry.
- Uniform solid right circular cone - The axis of a solid right circular cone is an axis of symmetry of the cone.
- Uniform solid hemisphere - The line the centre of the plane face perpendicular to the plane face is an axis of symmetry.
Centres of mass of solids of revolution
where y=f(x), and we have the limits of both integrals as x=a and x=b, where a and b are constants, and ρ is the mass per unit volume.
Centres of mass of surfaces of revolution
When part of a curve is rotated through 360 degrees about a fixed line we obtain a surface known as a surface of revolution. Examples include a hollow cone and a hemispherical shell. See M3 book for detailed diagrams and examples.
Simple cases of equilibrium of rigid bodies
A rigid body is in equilibrium if;
- The vector sum of the forces acting is zero, that is the sum of the components of the forces in any given direction is zero.
- The algebraic sum of the moments of the forces about any given point is zero.
Suspension of a body from a fixed point
- A rigid body hangs in equilibrium with its centre of mass vertically below the point of suspension.
Equilibrium of bodies on a horizontal plane
- If the line of action of the weight lies inside the area of contact, then the body is in equilibrium.
- If the line of action of the weight lies outside the area of contact, then it is not possible to have equilibrium as the reaction must act somewhere in the area of contact. In this case the body will topple.
- When the line of action of the weight passes through either end points of the area of contact. The body is in limiting equilibrium.
Equilibrium of bodies on an inclined plane
Same principles as previous section.
Slipping and toppling
- When the body is on the point of sliding, F = µR
- When the body is on the point of toppling, the reaction acts at the point about which the body will turn.
Standard Results for the Centre of Mass for Uniform Bodies
- Solid hemisphere, radius r -
from the centre
- Hemispherical shell, radius r -
from the centre
- Circular arc, radius, angle at centre 2x -
from centre
- Sector of circle, radius r, angle at centre 2x -
from centre
- Solid right circular cone, height h -
from the vertex
- Conical shell, height h -
from the vertex
Formulae
These standard results are given in the Exam in the Formula sheet. The Formula can be found at:
http://www.edexcel.org.uk/VirtualContent/83441/Formulae_book_Revised_Specification.pdf
Comments
Originally written by Widowmaker, and edited by VJ on TSR forums.
