Moment Equilibrium: For a rigid body to be in equilibrium, the sum of clockwise moments must equal the sum of anti-clockwise moments about any chosen pivot. The weight of the object must be included in this sum, using the distance from the pivot to the centre of mass.
Force Equilibrium: In addition to moments, the sum of all vertical forces must be zero. This typically involves the weight acting down and reaction forces from supports or tensions from strings acting upwards.
Pivot Selection: A strategic technique is to place the pivot at the location of an unknown force (like a support reaction). Since the distance from the pivot is zero, that force's moment becomes zero, allowing you to solve for other unknowns directly.
Definition of Tilting: Tilting occurs when an object begins to rotate about a pivot point because the moments are no longer balanced. This usually happens when a weight is moved toward one end of a supported beam.
The Point of Tilting: When an object is 'on the point of tilting' about a specific support, it is still technically in equilibrium, but the contact force at all other supports has dropped to exactly zero. This is a critical boundary condition used to solve limit-case problems.
Solving Tilting Problems: To find the maximum mass or distance before tilting, set the reaction force at the 'non-pivot' support to zero and solve the moment equation about the support that acts as the axis of rotation.
| Feature | Uniform Object | Non-Uniform Object |
|---|---|---|
| Density | Constant throughout | Varies by position |
| CoM Location | Geometric center/Symmetry lines | Shifted toward heavier regions |
| Weight Modeling | Acts at midpoint/centroid | Acts at a specified or calculated point |
| Complexity | Simple geometric calculation | Requires integration or experimental data |
Always Label the CoM: Before starting any calculation, explicitly mark the centre of mass on your diagram. If the rod is uniform, mark it at the midpoint; if non-uniform, use a variable like to represent its distance from a reference point.
Check for 'On the Point of Tilting': When you see this phrase, immediately identify which support the object is tilting around and set the reaction force at the other support(s) to zero. This is the most common 'hidden' information in mechanics problems.
Sanity Check: If you calculate the centre of mass of a uniform object and it is not at the geometric center, you have likely made an error in your moment equation or distance measurements.
Units and Consistency: Ensure all distances are measured from the same reference point (the pivot) and that units for mass (kg) are converted to weight (Newtons) by multiplying by .
