Uniform Laminas - Sectors and semi-circles

Understanding Uniform Laminas - Sectors and Semi-circles

• A uniform lamina is a thin flat object with uniform density and thickness.
• For further mechanics, we often focus on uniform laminas shaped like sectors of circles or semi-circles.

Centre of Mass for Uniform Semicircles

• The centre of mass of an object is the point where the whole mass of the uniform lamina is considered to be concentrated.
• For a uniform semi-circle, the centre of mass is located at a distance of 4r/3π from the flat edge along the axis of symmetry, where r is the radius of the semi-circle.

Centre of Mass for Uniform Circular Sectors

• A circular sector is the region enclosed by two radii and an arc of a circle.
• The centre of mass of a uniform circular sector lies along the axis of symmetry.
• For such a sector with radius r and angle θ in radians, the centre of mass from the centre of the circle to the centre of mass of the sector is given by 2r sin(θ/2) / 3θ.

Torque and Equilibrium

• In the context of mechanics, torque is the turning effect of a force.
• If a uniform lamina is in equilibrium, the sum of the torques about any point must be equal to zero.
• This can help solve many problems involving uniform laminas and balancing forces.

Solving Problems with Uniform Laminas

• When solving problems involving uniform laminas and forces, always start by sketching the problem setup.
• Identify where forces apply and the location of the centre of mass.
• Use the conditions for equilibrium, i.e., the sum of forces and torques equal to zero, to find unknowns.
• Throughout, be mindful of the units used and ensure consistency across calculations.