Forces
Learning Objective 1: Changes in Size and Shape
Forces are pushes or pulls that act on objects. When a force is applied to an object, it can produce changes in the object's size (lengthening or compressing) and shape (bending, twisting, or squashing).
Building on the concept of matter, forces cause the atoms within a solid to move slightly from their equilibrium positions. If the force is removed and the object returns to its original shape, it has undergone elastic deformation. If it remains permanently deformed, it has undergone plastic deformation.
Learning Objective 4: Newton's First Law (Inertia)
An object will remain at rest or continue in a straight line at constant speed unless acted on by a resultant force. This property is called inertia. Resultant Force: The single force that has the same effect as all the forces acting together.
If the resultant force is zero, the object's velocity does not change (it stays at rest or moves at constant speed).
If the resultant force is non-zero, the object accelerates.
Learning Objective 5: Changing Velocity
A resultant force changes the velocity of an object. Since velocity includes both speed and direction, a force can change:
- The speed (accelerate or decelerate).
- The direction of motion.
- Both speed and direction simultaneously.
Learning Objective 6, 7, 8: Friction and Drag
Friction is a force that opposes motion between surfaces or through fluids. Solid Friction: Acts between two solid surfaces in contact. It impedes motion and produces heating due to the work done against frictional forces. Drag (Fluid Friction): Acts on an object moving through a liquid or gas (e.g., air resistance). Drag increases as the speed of the object increases.
Learning Objective 11: Newton's Second Law (Supplement)
This law links force, mass, and acceleration. It states that the resultant force acting on an object is equal to the rate of change of its momentum. For constant mass, this simplifies to:
F = ma
Where:
F is the resultant force in newtons (N).
m is the mass in kilograms (kg).
a is the acceleration in metres per second squared (m/s^2).
Crucial Directionality: The force and the acceleration are always in the same direction. If the resultant force acts to the right, the acceleration is to the right.
Learning Objective 12: Circular Motion (Supplement)
An object moving in a circular path at constant speed is constantly changing direction. Therefore, it is accelerating (acceleration towards the centre). This requires a centripetal force acting perpendicular to the motion.
Qualitative relationships for an object of mass m moving with speed v in a circle of radius r under force F:
- If F increases (with m and r constant), the speed v increases.
- If F increases (with m and v constant), the radius r decreases (the circle becomes tighter).
- If m increases (with v and r constant), an increased force F is required to keep the object in the same circular path.
Learning Objective 9: Spring Constant (k)
The spring constant is defined as the force per unit extension required to stretch or compress a spring.
Equation:
k = \frac{F}{x}
Where:
k is the spring constant in newtons per metre (N/m) or newtons per centimetre (N/cm).
F is the force (load) applied in newtons (N).
x is the extension in metres (m) or centimetres (cm). Note: Extension is the change in length, not the total length.
Learning Objective 10: Limit of Proportionality
On a load-extension graph for an elastic solid, the limit of proportionality is the point beyond which the load is no longer directly proportional to the extension.
Below this point, the graph is a straight line through the origin (Hooke's Law applies).
Above this point, the graph curves.
- Note: Do not confuse this with the elastic limit, which is beyond the scope of this syllabus.
Step 1: Calculate Extension (x)
x = \text{Final Length} - \text{Original Length}
x = 14\ \text{cm} - 10\ \text{cm} = 4\ \text{cm}
Step 2: Calculate Spring Constant (k)
Using k = F/x:
k = \frac{5\ \text{N}}{4\ \text{cm}} = 1.25\ \text{N/cm}
If the question requires SI units (N/m):
x = 0.04\ \text{m}
k = \frac{5\ \text{N}}{0.04\ \text{m}} = 125\ \text{N/m}
A car of mass 1000 kg accelerates at 2\text{ m/s}^2. What is the resultant force?
Using F = ma:
F = 1000\text{ kg} \times 2\text{ m/s}^2 = 2000\text{ N}
The force acts in the same direction as the acceleration.
- Confusing Length with Extension
- Error: Using the total length of a spring in k = F/x. Correction: Always calculate extension first: x = L_{\text{final}} - L_{\text{original}}. The formula requires the change in length, not the absolute length.
- Ignoring Direction in Newton's Second Law
- Error: Stating only the magnitude of force or acceleration. Correction: Force and acceleration are vectors. You must state the direction (e.g., 'acceleration is 2\ \text{m/s}^2 to the right').
- Misidentifying Forces in Circular Motion
- Error: Thinking there is an outward 'centrifugal force' pushing the object out. Correction: There is only an inward centripetal force. The object wants to move in a straight line (inertia), but the force pulls it inward, causing circular motion.
- Moment Calculation Errors
- Error: Using the distance along the beam instead of the perpendicular distance from the pivot. Correction: Always use the perpendicular distance from the pivot to the line of action of the force.
Tip 1: Defining Spring Constant
Correct Usage Example: 'The force per unit extension.' or 'The force required to produce a unit extension.'
Why Examiners Accept This: Examiners look for the specific phrase 'force per unit extension'. Simply saying 'stiffness' is often insufficient unless defined. Do not forget to mention that it applies to an elastic spring.
Tip 2: Describing Friction/Drag
When asked to describe the effect of friction or drag.
'Friction opposes motion and produces heating.'
You must mention both the opposition to motion and the energy transfer (heating). Mentioning only one is often marked incomplete. Also, specify that drag acts on objects moving through fluids (liquids/gases).
Tip 3: Circular Motion Relationships
When describing how force affects circular motion.
'If the force increases, the radius decreases (for constant speed and mass).' or 'An increased mass requires an increased force to maintain the same speed and radius.'
Examiners accept specific causal statements. Use 'increases/decreases' clearly. Avoid vague terms like 'changes'. Ensure you specify which variables are held constant in your statement.
Learning Objective 13: Moment of a Force
The moment of a force is a measure of its turning effect.
Equation:
\text{Moment} = F \times d
Where:
F is the force in newtons (N).
d is the perpendicular distance from the pivot to the line of action of the force in metres (m) or centimetres (cm).
The unit of moment is the newton-metre (Nm) or newton-centimetre (Ncm).
Learning Objective 14: Principle of Moments
For an object to be in equilibrium:
- The resultant force must be zero (no linear acceleration).
- The resultant moment must be zero (no rotational acceleration).
This means: Sum of clockwise moments = Sum of anticlockwise moments about any pivot.
Learning Objective 16: Equilibrium
An object is in equilibrium when there is no resultant force AND no resultant moment acting on it. This implies the object is either at rest or moving with constant velocity (and not rotating).
Learning Objective 19, 20, 21: Centre of Gravity
- Centre of Gravity: The point through which the whole weight of an object appears to act. Experiment for Irregular Plane Lamina:
- Suspend the irregularly shaped plane lamina from a pivot point near its edge.
- Hang a plumb line (string with a weight) from the same pivot.
- Mark the position of the string on the lamina when it comes to rest.
- Repeat steps 1-3 for at least two other different pivot points.
- The centre of gravity is where the three lines intersect. Stability: An object is more stable if its centre of gravity is lower and its base is wider. If the line of action of the weight falls outside the base, the object will topple.
A uniform beam is balanced on a pivot at its centre. A 10 N weight is placed 20 cm to the left of the pivot. Where must a 5 N weight be placed to balance the beam?
Answer:
Using the principle of moments (Clockwise Moment = Anticlockwise Moment):
F_1 \times d_1 = F_2 \times d_2
10\ \text{N} \times 20\ \text{cm} = 5\ \text{N} \times d_2
200 = 5 \times d_2
d_2 = 40\ \text{cm}
The 5 N weight must be placed 40 cm to the right of the pivot.
Sample Question 2: Defining Limit of Proportionality
Define the term 'limit of proportionality' for a spring.
Answer:
The limit of proportionality is the point on a load-extension graph beyond which load is no longer directly proportional to extension. (Or: The point where the graph ceases to be a straight line through the origin.)
Sample Question 3: Load-Extension Graph
A student plots a load-extension graph for a spring. The graph is a straight line passing through the origin up to a load of 10 N, after which it curves. What does this indicate?
Answer:
The spring obeys Hooke's Law (load is proportional to extension) up to 10 N. The point at 10 N is the limit of proportionality. Beyond this point, the spring is no longer behaving elastically in a linear manner.
Sample Question 4: Determining Centre of Gravity
Describe how you would find the centre of gravity of an irregularly shaped piece of card.
Answer:
- Suspend the card from a pin at one point near the edge.
- Hang a plumb line from the same pin.
- Mark the line of the string on the card.
- Repeat for two other different suspension points.
- The centre of gravity is at the intersection of the three lines.
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