1 – Introduction to Vectors

2 – 1D Horizontal Motion

3 – Motion due to Gravity

4 – Continuous Change (Calculus)

5 – Forces and Systems

6 – Introduction to Energy and Momentum

7 – Impacts and Collisions

8 – Uniform Circular Motion

9 – Difference Equations

10 – Introduction to Graph Theory

11 – Path Optimisation

12 – MST Optimisation

13 – Project Optimisation

Mathematical Modelling Project

Revision Sheets

Past Papers (Higher Level)

HL Mock Exams

HL Exam Questions by Topic

OL Mock Exams

OL Exam Questions by Topic

Weekly Grinds

Forces and Systems

Forces and Systems

Key Points

A force \(\vec{F}\) is any interaction that can cause an object to accelerate.

Newton’s 1st Law: An object will remain at rest or moving at constant velocity unless a net external force acts upon it.

The mass \(m\) of an object is a measure of how much stuff is within that object.

Newton’s 2nd Law: When an object of mass \(m\) accelerates \(\vec{a}\) due to a net external force \(\vec{F}_{\text{net}}\) applied to it, the relationship between these three quantities is fully described by:

\[\vec{F}_{\text{net}} = m\vec{a}\]

The weight \(\vec{W}\) of an object of mass \(m\) is the gravitational force applied to that object and is given by

\[\vec{W}=m\vec{g}\]

where \(\vec{g}\) is the acceleration due to gravity.

Newton’s 3rd Law: For every action, there is an equal and opposite reaction.

A free body diagram consists only of an object, the force vectors acting on that object and the resulting acceleration vector of that object.

Tension \(\vec{T}\) is a pulling force caused by a string, rope or spring.

When an elastic string, rope or spring is stretched, its length is increased from its natural length \(l_0\) to a new length \(l\).

The magnitude of the tension for such a string is determined by using Hooke’s Law

\begin{align}|\vec{T}| = k(l-l_0)\end{align}

where the value of \(k\), known as the elastic constant, depends on the material that the string is made from.

The direction of the frictional force \(\vec{f}\) always acts to oppose the motion of the object.

An object of mass \(m\) is in contact with horizontal ground and subject to a force, or multiple forces, parallel to the ground, the sum of which is \(\vec{F}_{\parallel}\).

For a smooth surface, the situation is as below.

If the surface is instead rough, friction \(\vec{f}\) also acts on the object in a direction which opposes the object’s motion. 

The magnitude of \(\vec{f}\) is dependent upon whether \(|\vec{F}_{\parallel}|\) is greater or smaller than the magnitude of the limiting friction \(|\vec{f}_l| = \mu |\vec{R}|\).

  • If \(|\vec{F}_{\parallel}|\leq |\vec{f_l}|\), the object will not accelerate and \(|\vec{f}|=|\vec{F}_{\parallel}|\).
  • If \(|\vec{F}_{\parallel}| >|\vec{f_l}|\), the object will accelerate and \(|\vec{f}|=|\vec{f}_l|\).

This is as shown below where we gradually increase \(\vec{F}_\parallel\).

The work done \(W\) by a constant force \(\vec{F}\) in moving an object a displacement \(\vec{s}\) is given by

\[W = \vec{F} \cdot \vec{s}\]

The work done by a varying force \(\vec{F}(x)\) in moving an object in one dimension from \(x=a\) to \(x=b\) is given by
\[W = \int_a^b F(x)\,dx\]

where \(\vec{F}(x) = F(x) \vec{i}\).

If a constant amount of work \(W\) is being done by a force \(\vec{F}\) in a time \(t\), then the power \(P\) associated with that work done is defined as:

\[P = \frac{W}{t}\]

If a varying amount of work \(W\) is being done by a force \(\vec{F}\) in a time \(t\), then the power \(P\) associated with that work done is defined as:

\[P = \frac{dW}{dt}\]

If, at a time \(t\), an object moving with a velocity \(\vec{v}\) has a force \(\vec{F}\) applied to it, the power associated with that force is

\begin{align}P = \vec{F}\cdot\vec{v}\end{align}

If an entire system moves at the same acceleration rate, that system is said to have a common acceleration.

In order to relate the acceleration \(\vec{a}\) of an object to the net force \(\vec{F}_{\mbox{net}}\) on that object that is causing that acceleration, we perform the following algorithm:

  • Step 1: Draw a free body diagram.
  • Step 2: Write the force and acceleration vectors in component form.
  • Step 3: Use \(\vec{F}_{\text{net}}=m\vec{a}\).
  • Step 4: Insert any known coefficients and replace any similar coefficients.
  • Step 5: Dot the equation with unit vectors.

Causes of Confusion

A net force causes an object to accelerate. This does not mean that a net force must be applied to an object in order for it to move. As stated in Newton’s 1st Law, an object will also continue to move with a constant velocity if there is no net force on that object.

For example, an object floating in the depths of space can be moving without any forces being applied to it.

Likewise, an ice puck moving along perfectly smooth ice will continue moving indefinitely with a constant velocity until it hits the boundaries of the ice rink.

There are two other quantities that the concept of mass if often confused with:

  • Mass is not the same as volume. Volume instead describes the size of an object rather than how much stuff is in it. For example, a football and a bowling ball may have the same volume, but a bowling ball has significantly more mass.
  • Mass is not the same as weight. As we shall see shortly, weight is in fact a vector rather than a scalar and its value depends on how large the acceleration due to gravity is. For example, a bowling ball will have the same mass on both the Earth and the moon. Its weight, on the other hand, will be much less on the moon compared to on the Earth.

Typically, we use the following angle \(A\) when decomposing vectors.

In that case, based on the definitions of both sine and cosine, the magnitudes of the vectors are as follows

\begin{align}|\vec{a}_x| =|\vec{a}|\cos A \end{align}

\begin{align}|\vec{a}_y| =|\vec{a}|\sin A \end{align}

However, when decomposing vectors using rotated axes, we often instead know this angle

Again, based on the definitions of both sine and cosine, the magnitudes of the vectors are instead now given by

\begin{align}|\vec{a}_x| =|\vec{a}|\sin A \end{align}

\begin{align}|\vec{a}_y| =|\vec{a}|\cos A \end{align}

Exam Tips

In an exam question, if a surface is described as smooth, then we can assume that it is frictionless. Otherwise, it is said to be rough.

Typically, if we have two variables related by a fraction, we replace that fraction by a derivative when necessary.

If we instead have two variables related by a product, we typically replace that product by an integral when necessary.

If, we have one object with more forces acting on it than another, it is sensible to apply Newton’s 2nd Law to that object last.

We are more likely to make a mistake with our calculations with that object, and it is easier and quicker to correct such mistakes if they are found later in the solution.

We may assume that any system that contains multiple objects does accelerate (unless the question specifically states otherwise).

However, that we cannot make such an assumption if we instead have a system containing only one object.

If you are unsure about the direction of motion of an object (or a system), you can randomly choose a direction when drawing the free body diagram.

Once you find that object’s acceleration coefficient, the sign of that coefficient will tell you if that guess was correct or not. If it wasn’t, go back and redraw the correct acceleration vector.

Any time you are using rotated axes, it is a good idea to include those axes somewhere on any free body diagram that you draw.

This ensures that you can more easily visualise the vectors in relation to that axes and it also lets the examiner know that you are not using standard \(x\)-\(y\) axes.

When applying Newton’s second law to a system, we are free to use different axes for each object within that system.

Additional Resources

Forces and Motion: Basics – a collection of simulations showcasing net force, acceleration and friction

Hooke’s Law – visualise how much a spring stretches/contracts when a force is applied to it

Ramp: Forces and Motion – visualise the forces and motion of an object on an inclined plane