Physics (9702) | Turning effects of forces

Cambridge International AS & A Level Physics (9702) Study Notes

Welcome to the "Turning Effects of Forces" chapter! This section is all about understanding how forces don't just push or pull things linearly, but how they make objects spin, rotate, and balance. Mastering this concept is key to understanding everything from cranes to bicycles! Don't worry if it seems tricky; we'll break down moments, torque, and equilibrium into simple, manageable steps.

4.1 Centre of Gravity (CoG) and Moments

The Centre of Gravity (CoG)

In physics calculations, objects often have complex shapes, but their weight is the total gravitational force acting on all their tiny particles. We need a single, simple point to represent where this overall weight acts.

Definition of Centre of Gravity (CoG)

The Centre of Gravity (CoG) of an object is the single point where the entire weight of the object may be considered to act.

• For a uniform object (where mass is evenly distributed) like a symmetric rod or sphere, the CoG is exactly at the geometric centre.
• For non-uniform or irregularly shaped objects, the CoG might not be the geometric centre.

Example: When you hold a hammer, the CoG is located much closer to the heavy head than to the handle. If you balance the hammer on your finger, the balance point is the CoG.

Moments of a Force

A force causes linear motion, but a moment is the measure of the turning effect produced by a force. It’s what allows you to turn a wrench or push open a heavy door.

Definition and Calculation of Moment

The moment of a force about a point (the pivot) is defined as the product of the force and the perpendicular distance from the pivot to the line of action of the force.

Formula:
\( \text{Moment} = \text{Force} \times \text{Perpendicular distance} \)
\( M = F \times d \)

• Units: Since Force is in Newtons (N) and distance is in metres (m), the unit for the moment is the Newton metre (N m).
• The distance \(d\) must be the perpendicular distance from the pivot to the line along which the force acts.

Key Concept Alert: If the force acts directly through the pivot, the perpendicular distance is zero (\(d=0\)), so the moment is zero. The force produces no turning effect!

Analogy: Why do door handles work best when placed far away from the hinges? Because maximizing the perpendicular distance (\(d\)) means you need a smaller force (\(F\)) to achieve the necessary moment (\(M\)).

Quick Review: Moments

A moment describes rotation around a single fixed point (the pivot).

4.2 Couples and Torque

Sometimes, rotation isn't caused by a single force around a pivot, but by a pair of forces acting together. This special arrangement is called a couple.

What is a Couple?

A couple is a pair of forces that acts to produce rotation only.

A couple must satisfy three conditions:

1. The two forces must be equal in magnitude.
2. They must be parallel to each other.
3. They must act in opposite directions.

When a couple acts on an object, the resultant linear force on the object is zero (since the forces are equal and opposite), meaning there is no linear acceleration. However, they create a pure turning effect!

Example: Using both hands to turn a steering wheel, or twisting a screwdriver.

Torque of a Couple

The turning effect produced by a couple is called the torque (\(\tau\)).

Definition and Calculation of Torque

The torque of a couple is defined as the product of one of the forces and the perpendicular distance between the lines of action of the two forces.

Formula:
\( \tau = F \times d \)

• Here, \(F\) is the magnitude of one of the forces, and \(d\) is the perpendicular separation between the two forces.
• Units: The unit for torque is also the Newton metre (N m).

4.3 Equilibrium of Forces

If an object is stationary or moving at a constant velocity, it is said to be in equilibrium. This requires both linear and rotational balance.

Conditions for Equilibrium (Syllabus 4.2.2)

For a system to be in equilibrium, there must be no resultant force and no resultant torque.

Condition 1: Translational Equilibrium (No Resultant Force)

The object is not accelerating linearly.

\( \Sigma F = 0 \)

This means the vector sum of all forces acting on the object must be zero. Generally, this is applied by ensuring:

• Sum of forces acting up = Sum of forces acting down.
• Sum of forces acting left = Sum of forces acting right.

Condition 2: Rotational Equilibrium (No Resultant Torque)

The object is not accelerating angularly (not starting to spin).

\( \Sigma \tau = 0 \)

This leads directly to the Principle of Moments.

The Principle of Moments (Syllabus 4.2.1)

If an object is in rotational equilibrium, the sum of the moments causing clockwise rotation about any point must be equal to the sum of the moments causing anticlockwise rotation about the same point.

Mathematically:
\( \Sigma \text{Clockwise Moments} = \Sigma \text{Anticlockwise Moments} \)

Step-by-Step Application of the Principle of Moments

This principle is the cornerstone of solving balancing problems involving beams, planks, and levers.

1. Draw a Diagram: Include all forces (weight, reactions, applied forces) and mark the relevant distances.
2. Choose a Pivot: You can choose *any* point as the pivot, but choosing the point where an unknown force acts is usually smart, as that force's moment becomes zero (since \(d=0\)), simplifying the calculation.
3. Identify Rotation Direction: Determine whether each force causes a clockwise (CW) or anticlockwise (CCW) moment about your chosen pivot.
4. Calculate Moments: Use \(M = F \times d\) for every force.
5. Apply the Principle: Set the sum of CW moments equal to the sum of CCW moments and solve for the unknown quantity.

Common Mistake to Avoid: Always remember to include the weight of the beam/rod itself, which acts downwards from the Centre of Gravity of the beam (often its midpoint). If the question says the beam is "light," you can ignore its weight.

Key Takeaway: Equilibrium

To solve any equilibrium problem, you usually need to apply both conditions: zero resultant force (linear balance) and zero resultant moment (rotational balance).

4.4 Representing Coplanar Forces in Equilibrium (Vector Triangles)

When an object is held in equilibrium by three coplanar forces (forces acting in the same plane), we can use a simple geometric method to find unknown forces or angles.

The Rule of Three Forces (Syllabus 4.2.3)

If an object is in equilibrium under the action of three forces, two things must be true:

1. The lines of action of the three forces must intersect at a single point (unless the forces are all parallel).
2. The three forces must form a closed loop when drawn head-to-tail.

Using a Vector Triangle

Since the resultant force (\(\Sigma F\)) must be zero for equilibrium, when we add the three forces together using the head-to-tail method, the end of the last vector must land exactly on the start of the first vector. This forms a closed vector triangle.

Process:

1. Choose a suitable scale (e.g., 1 cm = 10 N).
2. Draw the first known force vector.
3. From the tip (head) of the first vector, draw the tail of the second vector, ensuring the direction is correct.
4. The third force vector must connect the tip of the second vector back to the tail of the first vector (closing the triangle).
5. You can then use measuring tools (protractor and ruler) or trigonometry (Sine Rule, Cosine Rule) to determine the magnitude and direction of the unknown forces.

Did you know? This technique is particularly useful for solving problems involving objects hanging from strings or supported by inclined rods, where calculating moments might be difficult due to complex angles.

Quick Review: Vector Triangles

A vector triangle is used when three coplanar forces maintain equilibrium. The three vectors, when drawn head-to-tail, must form a closed loop, confirming that the net resultant force is zero.