Tilting (AQA A-Level Mathematics): Revision Notes

• Moment (Torque): A moment is the turning effect of a force about a pivot point. It is calculated as:

$\text{Moment} = \text{Force} \times \text{Perpendicular Distance from the Pivot}$

Moments cause rotation and can either resist or promote tilting.

• Centre of Mass: The point where the entire weight of the body acts. The position of the centre of mass relative to the base of support is crucial in determining whether an object will tilt.
• Normal Force: The support force exerted by a surface to hold an object against gravity. When tilting occurs, the normal force at one edge may decrease to zero as the object begins to rotate around that edge.

Step 1: Identify Forces and Moments

• Weight ( $mg$ ): Acts vertically downward through the centre of mass, located at the midpoint of the box's width.
• Tilting Point: Assume the box begins to tilt about the edge of its base. Step 2: Calculate the Moment Caused by the Applied Force

The moment about the pivot (the edge of the base) due to the applied force $F$ is:

$\text{Moment of } F = F \times h$

Step 3: Calculate the Moment Caused by the Weight

The moment about the same pivot due to the weight $mg$ is:

$\text{Moment of } mg = mg \times \frac{w}{2}$

Here, $\frac{w}{2}$ is the horizontal distance from the pivot point to the centre of mass.

Step 4: Condition for Tilting

Tilting occurs when the moment due to the applied force equals or exceeds the moment due to the weight:

$F \times h \geq mg \times \frac{w}{2}$

Solving for $F$ :

$F \geq \frac{mg \times \frac{w}{2}}{h} = \frac{mgw}{2h}$

This equation provides the minimum force required to cause the box to begin tilting.

Problem: A $10$ $kg$ box with a height of $0.5$ $m$ and a width of $0.3$ $m$ is on a horizontal surface. What is the minimum horizontal force that must be applied at the top of the box to cause it to tilt?

Tilting in mechanics is a situation where an object begins to rotate about one of its edges due to an unbalanced moment caused by external forces. By analysing the moments around the pivot point, you can determine the conditions under which tilting occurs and calculate the force needed to initiate tilting. This concept is important in ensuring the stability of objects and structures in various practical situations.