# Mathematics Extension 2 – Mechanics

Mechanics

## Circular Motion

Circular motion describes motion of a particle around a circle. Circular motion can occur in various different ways, such as:

• A particle moves freely in a horizontal circle with no strings attached.
• A particle moves in a horizontal circle in contact with a smooth surface.
• A particle moves in a horizontal circle that is joined by a taut string/rod to a fixed point on a smooth surface.
• A particle moves in a horizontal circle inside a smooth hemispherical bowl.

Let’s now look at some terminology, formulas and forces that are used to analyse circular motion.

### Angular Velocity

Q is a fixed point and P is a moving point. PQ subtends an angle $Latex formula$ at the centre of a circle with radius $Latex formula$. The angular velocity $Latex formula$ (omega) is the rate of change of $Latex formula$(measured in radians) with respect to time $Latex formula$, i.e.

 $Latex formula$

Angular velocity is measured in radians per second.

In uniform circular motion, constant angular velocity can also be defined as

 $Latex formula$

Where, $Latex formula$

$Latex formula$ $Latex formula$

### Angular Acceleration

Angular acceleration is the rate of change of angular velocity with respect to time, i.e.

 $Latex formula$

### Linear Velocity

The linear velocity $Latex formula$, also referred to as the tangential velocity or the instantaneous velocity of a point moving in circular motion.Consider, the rate of change of the arc length $Latex formula$ represents the tangential velocity $Latex formula$.

$Latex formula$

$Latex formula$

$Latex formula$

 $Latex formula$

### Forces Acting on a Particle Moving in a Circle

The Mathematics Extension 2 syllabus requires students to be able to prove the tangential and normal components of the force acting on a particle moving in a circle of radius $Latex formula$, with angular velocity $Latex formula$. However, motion is restricted to uniform circular motion, i.e. constant angular velocity.

#### The tangential force on a particle moving in non-uniform circular motion

Consider,

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Now by Newton’s 2nd Law, we can derive the formula:

 $Latex formula$

#### The normal force on a particle moving in non-uniform circular motion

Consider the variable point $Latex formula$ which makes an angle $Latex formula$ with the positive $Latex formula$-axis.

Now, applying implicit differentiation with respect to time

 $Latex formula$$Latex formula$$Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ Since we are considering uniform circular motion, then$Latex formula$. $Latex formula$ $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Since, $Latex formula$

 $Latex formula$

Now by Newton’s 2nd Law, we can derive the formula:

 $Latex formula$

Now, by substituting $Latex formula$, we have

 $Latex formula$

Note: the normal force $Latex formula$ is also called the centripetal force.

The next examples demonstrate some of the concepts we have looked at so far.

### Example 1

A particle of mass 6 kg is travelling at constant velocity of 2$Latex formula$ round a circle of radius$Latex formula$ m. Find the rate at which the angle at the centre of the circle is changing if the centripetal force has a magnitude of 12 Newtons.

### Solution 1

$Latex formula$

$Latex formula$

$Latex formula$
$Latex formula$m
We also have $Latex formula$

$Latex formula$
$Latex formula$ rad/s

### Reaction Forces

Reaction forces are a result of Newton’s third law which states for every action there’s an equal and opposite reaction force. When a particle P is in contact with a surface, the force acting on P by the surface is called the reaction force. The reaction force exerts on a particle is always at right angle to the surface.

### Tension Forces

Tension forces are also a result of Newton’s third law. When a particle is attached to a taut string, the string exerts a tension force on the particle. Also, when a particle attached to a taut string is moving in circular motion, the tension force is the centripetal force. Tension forces are of greater significance in the conical pendulum section.

### Example 2

A particle of mass $Latex formula$ kg is attached to a taut string of length $Latex formula$, which is fixed on the table. The particle moves in a horizontal circle in contact with the smooth table. If the maximum tension that the string can support without breaking is equal to a gravitational weight force on an object of $Latex formula$ kg, find the maximum number of revolutions per second that the particle can make without breaking the string.

### Solution 2

The tension force is equal to the centripetal force.

$Latex formula$

We are given that the maximum tension is equal to the gravitational weight force on mass $Latex formula$

$Latex formula$

$Latex formula$ rad/s

### Example 3

A particle moves with constant angular velocity $Latex formula$ in a horizontal circle of radius $Latex formula$ on the inside of a fixed smooth hemispherical bowl of internal radius $Latex formula$. Show that $Latex formula$.

### Solution 3

$Latex formula$

$Latex formula$

Resolving forces vertically:

$Latex formula$

$Latex formula$

Resolving forces horizontally:

$Latex formula$

$Latex formula$

### Example 4

[HSC 2007, Question 3]

A particle $Latex formula$ of mass $Latex formula$ undergoes uniform circular motion with angular velocity $Latex formula$ in a horizontal circle of radius $Latex formula$ about $Latex formula$. It is acted on by the force due to gravity, $Latex formula$, a force $Latex formula$ directed at an angle $Latex formula$ above the horizontal and a force $Latex formula$ which is perpendicular to $Latex formula$, as shown in the diagram.

(i) By resolving forces horizontally and vertically, show that

$Latex formula$

(ii) For what values of $Latex formula$ is $Latex formula$?

### Solution 4

i)

Resolving forces horizontally:

$Latex formula$

Multiply by $Latex formula$,

$Latex formula$

Resolving forces vertically:

$Latex formula$

Multiply by $Latex formula$,

$Latex formula$

Solving simultaneously, we have:

$Latex formula$

$Latex formula$

ii)

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$ rad/s

## The Conical Pendulum

The Conical Pendulum describes the motion of a particle moving in (uniform) circular motion which is suspended from a fixed point by a light taut string. There are different forms of conical pendulum, for example:

• A particle is suspended from a fixed point by a light inextensible string.
• A particle is suspended from a fixed point by a light inextensible string. The particle is moving in uniform circular motion on a smooth horizontal table.
• A string passes through a smooth hole in a smooth table where each end of the string is attached to a particle. One of the particles is on the table moving in uniform circular motion and the other particle is below the table where its motion can be described by the motion of a conical pendulum.

When dealing with conical pendulum questions, students need to be able to:

(i) Understand all the different forces acting on the particle, e.g. tension force, centripetal force, gravitational force, normal force and reaction force.

(ii) Resolve forces in vertical and horizontal directions separately.

(iii) Solve the question by methods of substitution and elimination.

### Example 5

[HSC 2012, Question 7]

A particle $Latex formula$ of mass $Latex formula$ attached to a string is rotating in a circle of radius $Latex formula$ on a smooth horizontal surface. The particle is moving with constant angular velocity $Latex formula$. The string makes an angle $Latex formula$ with the vertical. The forces acting on $Latex formula$ are the tension $Latex formula$ in the string, a reaction force $Latex formula$ normal to the surface and the gravitational force $Latex formula$.

Which of the following is the correct resolution of the forces on $Latex formula$ in the vertical and horizontal directions?

(A) $Latex formula$ and $Latex formula$

(B) $Latex formula$ and $Latex formula$

(C) $Latex formula$ and $Latex formula$

(D) $Latex formula$ and $Latex formula$

### Example 6

[HSC 2008, Question 3]

A particle $Latex formula$ of mass $Latex formula$ is attached by a string of length $Latex formula$ to a point $Latex formula$. The particle moves with constant angular velocity $Latex formula$ in a horizontal circle with centre $Latex formula$ which lies directly below $Latex formula$. The angle the string makes with $Latex formula$ is $Latex formula$.

The forces acting on the particle are the tension, $Latex formula$, in the string and the force due to gravity, $Latex formula$.

By resolving the forces acting on the particle in the horizontal and vertical directions, show that

$Latex formula$

### Solution 6

Resolving forces horizontally:

$Latex formula$

Resolving forces vertically:

$Latex formula$

Solving these simultaneously, we have:

$Latex formula$

We also know that $Latex formula$.

$Latex formula$

$Latex formula$

### Example 7

[HSC 2011, Question 5]

A small bead of mass $Latex formula$ is attached to one end of a light string of length $Latex formula$. The other end of the string is fixed at height $Latex formula$ above the centre of a sphere of radius $Latex formula$, as shown in the diagram. The bead moves in a circle of radius $Latex formula$ on the surface of the sphere and has constant angular velocity $Latex formula$. The string makes an angle of $Latex formula$ with the vertical.

Three forces act on the bead: the tension force $Latex formula$ of the string, the normal reaction force $Latex formula$ to the surface of the sphere, and the gravitational force $Latex formula$.

(i) By resolving the forces horizontally and vertically on a diagram, show that

$Latex formula$

and

$Latex formula$

(ii) Show that

$Latex formula$

(iii) Show that the bead remains in contact with the sphere if $Latex formula$.

### Solution 7

i)

Resolving forces horizontally:

Note that the centripetal force is directed towards the centre (and is the net force). From the diagram above, the force F is contributing positively whereas the force N is opposing the overall force towards the centre. (It may help to draw a line perpendicular to the net force direction, which in this case is a vertical line. Any force to the left of the vertical line in this case would contribute positively to the centripetal force, whereas any force to the right would be negative).

$Latex formula$

Resolving forces vertically:

Since the bead is not moving up or down, the net force vertically is 0. Again, the line perpendicular to the vertical motion is a horizontal line, and clearly, F and N are above the horizontal whereas the weight force is below.

$Latex formula$

ii)

We need to eliminate F.

Consider

$Latex formula$

Multiply both sides by $Latex formula$ and make $Latex formula$ the subject.

$Latex formula$

Similarly, for the other equation, multiplying by sin $Latex formula$,

$Latex formula$

Now consider (1) – (2)

$Latex formula$($Latex formula$

$Latex formula$

Divide both sides by $Latex formula$

$Latex formula$

iii)

The bead remains in contact when $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Note that $Latex formula$ and that $Latex formula$.

$Latex formula$

$Latex formula$

Substituting this in, we have:

$Latex formula$

$Latex formula$

### Example 8

[HSC 2009, Question 4]

A light string is attached to the vertex of a smooth vertical cone. A particle $Latex formula$ of mass $Latex formula$ is attached to the string as shown in the diagram. The particle remains in contact with the cone and rotates with constant angular velocity $Latex formula$ on a circle of radius $Latex formula$. The string and the surface of the cone make an angle of $Latex formula$ with the vertical, as shown.

The forces acting on the particle are the tension, $Latex formula$, in the string, the normal reaction, $Latex formula$, to the cone and the gravitational force $Latex formula$.

(i) Resolve the forces on $Latex formula$ in the horizontal and vertical directions.

(ii) Show that $Latex formula$ and find a similar expression for $Latex formula$.

(iii) Show that if $Latex formula$ then

$Latex formula$

(iv) For which values of $Latex formula$ can the particle rotate so that $Latex formula$?

### Solution 8

i)

Resolving forces horizontally:

$Latex formula$

Resolving forces vertically:

$Latex formula$

ii)

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

iii)

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

iv)

Note that $Latex formula$ and $Latex formula$

Therefore, $Latex formula$.

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

### Example 9

[HSC 2006, Question 5]

A particle, $Latex formula$, of mass $Latex formula$ is attached by two strings, each of length $Latex formula$, to two fixed points, $Latex formula$ and $Latex formula$, which lie on a vertical line as shown in the diagram.

The system revolves with constant angular velocity $Latex formula$ about $Latex formula$. The string $Latex formula$ makes an angle $Latex formula$ with the vertical. The tension in the string $Latex formula$ is $Latex formula$ and the tension in the string $Latex formula$ is $Latex formula$ where $Latex formula$ and $Latex formula$. The particle is also subject to a downward force, $Latex formula$, due to gravity.

(i) Resolve the forces on $Latex formula$ in the horizontal and vertical directions.

(ii) If $Latex formula$, find the value of $Latex formula$ in terms of $Latex formula$ and $Latex formula$.

### Solution 9

i)

Resolving forces horizontally:

$Latex formula$

Resolving forces vertically:

$Latex formula$

ii)

Substitute $Latex formula$

$Latex formula$

$Latex formula$

Consider $Latex formula$,

$Latex formula$

$Latex formula$

## Motion around a Circular Banked Track

In this section, we will study the forces of a particle travelling at constant velocity around a banked circular track. Since the track is banked at an angle to the horizontal to minimise the risk of slipping, the force of friction between the particle and the track needs to be considered. Optimum speed occurs when the friction force equals to zero. When dealing with frictional forces, it is required to direct them in the opposite direction to the sliding movement of the particle, in order for the particle to not slip. Let’s now derive the following results as required by the syllabus.

$Latex formula$

Now analyse the forces on a particle of mass traveling at constant velocity around a horizontal circular arc with radius of curvature on a track that is inclined at an angle to the horizontal, as shown below.

### Example 10

[HSC 2005, Question 3]

The diagram shows the forces acting on a point $Latex formula$ which is moving on a frictionless banked circular track. The point $Latex formula$ has mass $Latex formula$ and is moving in a horizontal circle of radius $Latex formula$ with uniform speed $Latex formula$. The track is inclined at an angle $Latex formula$ to the horizontal. The point experiences a normal reaction force $Latex formula$ from the track and a vertical force of magnitude $Latex formula$ due to gravity, so that the net force on the particle is a force of magnitude $Latex formula$ directed towards the centre of the horizontal circle.

By resolving $Latex formula$ in the horizontal and vertical directions, show that

$Latex formula$

### Solution 10

Resolving forces horizontally:

$Latex formula$

Resolving forces vertically:

$Latex formula$

Consider $Latex formula$

$Latex formula$

Also consider $Latex formula$,

$Latex formula$

Using a triangle, we see that:

$Latex formula$

Substituting this into the above we have:

$Latex formula$
$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

### Example 11

[HSC 2010, Question 4]

A bend in a highway is part of a circle of radius $Latex formula$, centre $Latex formula$. Around the bend the highway is banked at an angle $Latex formula$ to the horizontal.

A car is travelling around the bend at a constant speed $Latex formula$. Assume that the car is represented by a point $Latex formula$ of mass $Latex formula$. The forces acting on the car are a lateral force $Latex formula$, the gravitational force $Latex formula$ and a normal reaction $Latex formula$ to the road, as shown in the diagram.

(i) By resolving forces, show that $Latex formula$.

(ii) Find an expression for $Latex formula$ such that the lateral force $Latex formula$ is zero.

### Solution 11

i)

Resolving forces horizontally:

$Latex formula$

Resolving forces vertically:

$Latex formula$

Consider $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

ii)

If lateral force force $Latex formula$ is zero, then $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$