Solve sliding/toppling problems

Solving Sliding/Toppling Problems

Sliding and toppling problems refer to scenarios dealing with equilibrium and stability. A body is in equilibrium when it is at rest and there are no resultant forces or moments.
To solve such problems, understanding and applying the principles of Centre of Mass (CoM) is crucial.

Key Principles

An object will begin to slide when the frictional force between the object and the surface it is resting on is exceeded by an externally applied force.
The limiting frictional force (F) can be calculated as F = μR where μ is the coefficient of friction between surfaces and R is the normal reaction force.
A body will topple over when the line of action of the gravity acting through the CoM moves outside the base of the body. In that case, there is a turning effect that causes the body to rotate.

Steps to Solve Sliding/Toppling Problems

Start by drawing a diagram to represent the problem and identifying all the forces acting on the body.
Mark the CoM and draw the line of action of weight (which is the force of gravity acting through the CoM).
Split the system into components, if required.
Use Newton’s laws to construct equations relating to the forces involved. The resolution of forces method can be useful here.
- Newton’s First Law: Objects at rest or in a uniform motion will continue to be so unless acted upon by an external force.
- Newton’s Second Law: The force on an object is equal to its mass times its acceleration.
- Newton’s Third Law: For every action, there is an equal and opposite reaction.
If the problem involves toppling, calculate the moments about the pivot point and equate it to zero (since in equilibrium, total clockwise moment = total counter-clockwise moment). The equation for the moment is moment = force × distance.
For sliding issues, compare the forces trying to move the object with the limiting frictional force. Remember, the object moves if the force is greater than limiting friction.
Finally, solve the equations simultaneously, if multiple equations are involved.

Tips and Precautions

Be cautious to choose the correct pivot point when considering a toppling problem.
The shape and incline of the surfaces where the body is resting can affect the likelihood of sliding or toppling. These factors should be considered in your calculations.
Problems can be more complicated with composite bodies. Always decompose them into simpler parts.
Confirmation that a body will slide or topple is highly dependent on the values of the coefficient of friction (μ) and the angle of inclination (θ).
Always check your calculations. A common slide/topple problem mistake is incorrectly calculating the acting forces or moments.
Make use of Pythagoras’ theorem and trigonometry when dealing with inclines or complicated problem geometries. Remember, sinθ = Opposite/Hypotenuse, cosθ = Adjacent/Hypotenuse, and tanθ = Opposite/Adjacent.

Understanding and applying these steps and key principles to solve sliding and toppling problems will be a useful skill not just for assessing Centre of Mass scenarios, but also for handling real-world physics and engineering problems.