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Mechanics Revision Revision for Topic 2- Mechanics for the IB Diploma

Jake for IB Screwed

Topic 2

Mechanics

What is Mechanics?

Mechanics is the study of motion and the formulation of equations to describe this motion.

It is the most basic of all fields in Physics but is the one that is required in all other fields.

In the IB diploma it is split into four regions Kinematics, Forces, Energy, and Circular Motion,

with each subtopic being quite diverse and informative in its own right. However for success

in IB Physics a firm grasp of mechanics is required as it can pop up in almost any other topic.

In writing this I assume you have a working understanding of Vectors and Scalars already

although if you do not another document will be created soon covering everything from

what are vectors through to adding and other techniques not required for the IB Diploma

but that are required for HL Mathematics.

If there is anything you feel could be written better or explained more please send an email

and I will try to work on it for you.

For any further Help with Mechanics or anything else in physics I recommend the YouTube

Channel, Khan Academy found at http://www.youtube.com/user/khanacademy. It has

everything from maths to physics to chemistry and is excellent to explain things to you.

Questions are at the end of the document.

Subtopic 1- Kinematics

What is kinematics?

Kinematics is the study of motion and how motion can be predicted. Within the IB course kinematics

simply involves the study of motion resulting from a constant acceleration. By the end of yr 12 you

should be able to derive the equations of motion using calculus however this approach is NOT

necessary for the IB Diploma

Definitions

Equations of Motion- The set of four equations in which can be used (when acceleration is constant)

to predict the motion of an object

Scalar- Simply a number

Vector- A scalar attached to a direction (50 metres north)

Distance- A scalar quantity representing the length travelled. An example is the distance from

Brisbane to Sydney. It is path dependent in that the distance depends on the path you take. Distance

cannot be negative.

Displacement- A vector representing the distance travelled. However it is the distance from a

specific point and the direction that this distance is in. It is path independent in that how you move

between the initial point and the secondary point does not matter.

Speed- speed is a scalar quantity representing the distance traversed in a certain amount of time. It

can be thought of as being what you see on a speedometer in a car, in that there is no direction

prescribed to the speed you read off the speedometer. Speed cannot be negative

Velocity-Velocity is simply speed with a direction attached.

Acceleration- The rate of change of velocity/speed with respect to time. It is either a vector or a

scalar quantity. It can be thought of as how in a car you press the accelerator and the car speeds up

and then if you press the brake, you slow down. Both of these are examples of acceleration with one

being positive and the other negative.

The equations of Motion

𝑣 = 𝑢 + 𝑎𝑡

𝑠 =𝑢 + 𝑣

2𝑡

𝑠 = 𝑢𝑡 +1

2𝑎𝑡2

𝑣2 = 𝑢2 + 2𝑎𝑠

v=final velocity

u= initial velocity (at time zero)

s= displacement

a=acceleration

t= time

Notes about equations.

These equations only hold if the acceleration is constant.

Graphical representation of Motion

For a graph of displacement against time the slope at a point on the graph will give the velocity at

that point.

For a graph of velocity against time, the area below the graph between two times t1 and t2 will give

the displacement of the object between these times. The slope of this graph at a point will give the

acceleration at this point.

For an acceleration-time graph the area below the graph between two times will give the change in

velocity between those two points.

These above allow for with the knowledge of calculus for one to derive the equations of motion.

Subtopic 2- Forces

What is involved in Forces?

Forces involves the application of Newtons Laws to determine motion.

Definitions

Force- A force can be understood as a push or a pull that one object exerts on another.

Force is a vector.

Momentum- momentum can be understood as the quantity that says how hard it is to stop

the motion of an object. Momentum is equal to the product of the mass of the object and

the velocity of the object. Momentum is always conserved. This means that in a closed

system (somewhere where nothing can escape) that the momentum at any time will always

be the same. Momentum is a vector

Rest- Motionless

State of motion- The way in which an object is moving, either rest or at 20km/hr etc.

Newtons Law’s of Motion

First Law-

A body will stay in its state of motion unless an unbalanced force acts upon it.

This means that an object will continue moving in the same direction at the same speed, (or

stay at rest) if there is no UNBALANCED force acting upon it. This does not mean that there

can’t be a force acting upon it just that this force must be balanced. This can be thought of if

you hold a 1kg weight up. Gravity will cause a force upon the weight which you balance with

an upward force thereby making the weight not move. The hardest part with this law is that

in nature we see things slow down, which is actually due to the force imparted by friction.

You must also need to recognise when you need to use it which occurs in questions with

words like at rest, constant velocity, and state of motion.

Second Law

𝐹 = 𝑚𝑎 =∆𝑝

∆𝑡

F-Force

m=mass

a-acceleration

p- momentum

t- time

These can be used as to allow for the conversion from force to allow for the utilisation of

the equations of motion to allow for the prediction of motion due to a force.

Third Law

To every action there is always an equal and opposite reaction: or the forces of two bodies on each

other are always equal and are directed in opposite directions.

This means that if one object, A, subjects another, B, to a force then B will subject a force of

equal magnitude but opposite direction to object B. As the forces are acting on different

bodies the forces do not cancel. This can be visualised as you pushing on a wall. If no force

was imparted on you then you would move through the wall because you would have an

unbalanced force upon you.

Impulse

Impulse is the change in momentum due to a force exerted over a specific amount of time.

It is usually encountered in the form of Force Time graphs in which you are required to find

the area under the graph and find the change in momentum due to a certain force. It is also

used to show how a low force over a long period of time can have as much effect as a large

force over a short period of time.

For calculations the Second Law modified to the form of ∆𝑝 = 𝐹∆𝑡 is used.

Conservation of Momentum

The conservation of momentum says that in a closed system, the momentum of the system

is constant.

This can be used to calculate the results of collisions between particles if one has enough

information about the before and after states of a collision.

For calculations you put all the momentums before the collision on the left side of the

equation, and all the momentums after the collision on the right and solve for the

unknowns. For a numerical solve of a variable, there must only be one variable in the

equation.

Normal force

The normal force is the force that is exerted on a body by something it is resting on. The

magnitude is equal to the magnitude of the force of the object into the surface and the

direction is perpendicular to the orientation of the surface (at 90 degrees).

Tension

Tension is the force that a string/rope applies to an object. This often is used in pulley

questions which require one to find the resultant motion on the objects

Subtopic 3- Energy and Work

What is energy?

Energy is the fundamental quantity in which reflects how much potential something has to

do or how much has been done on a system.

Energy is separated into two main quantities at this level.

Potential- The potential that an object has to turn into Kinetic Energy.

Kinetic- Reflects the motion that the object has.

Work is the change in one of these quantities. A certain amount of work will increase the

Kinetic Energy by that amount and decrease the potential energy by that amount or vice

versa. Work can be either negative or positive. Kinetic Energy can only be positive but in

some circumstances potential energy can be negative. The total Energy of a closed system is

constant, but the values of kinetic energy and potential energy can change.

Kinetic Energy

The kinetic energy of an object can be found through either of these expressions

𝐸𝑘 =1

2𝑚𝑣2 ,

𝑝2

2𝑚

The subscript after E signifies it is kinetic Energy.

Kinetic Energy is conserved in some collisions which are called elastic collisions. These can

be visualised like pool balls clashing into each other. In these collisions the Kinetic Energy

before the collision and after the collision will be the same and so allows for one to solve

simultaneously in cases with more than two unknowns by also applying the conservation of

momentum.

Cases where the conservation of kinetic Energy do not apply are called inelastic collisions

and examples of these are explosions, separation or combination of particles (car crashes)

and any collision where there is some “stickyness” between the particles. All collisions are

slightly inelastic.

Work

Work is the change in Kinetic or Potential Energy. In the case of a constant force in a

direction it is calculated by multiplying the Force in the direction of the displacement by the

displacement. This is written as𝑊 = 𝐹𝑠 𝑐𝑜𝑠𝜃, where theta is the angle between the Force

and the direction of displacement. Work is path independent in that the work moving from

one place to another is the same no matter what path they take to get there.

Work can also be found as the area below a Force displacement graph.

Potential Energy

Potential Energy is the potential energy that could be turned into kinetic energy that a

particle has. It can be calculated by finding the work done moving a particle from infinitely

far away to the point in question but this is often infeasible and is not expected to be able to

be done by the IB, and as such the equations for potential energy are given when needed.

None are needed for Topic 2 though so they will be discussed when encountered. All that is

needed to be known is that when work is done it will either be added to the potential

energy or taken away and vice versa for Kinetic Energy.

Applying this if you know at the top of a hill the Potential Energy is X and at the bottom of

the hill it is Y, then the kinetic energy of an object dropped down that hill will be

𝐸𝑘 = 𝑋 − 𝑌 assuming that the initial kinetic energy of the object was zero.

Subtopic 4- Circular Motion

What is Circular Motion?

Circular motion is the application of mechanics to systems where there is an object moving

in a circular path.

Firstly for an object to be undergoing circular motion the speed of the object must be

constant but its velocity must always be changing. Secondly the acceleration of the object

must always point perpendicularly to the direction of motion at any instant in time.

For circular motion the equations of motion take the form

𝑣 =2𝜋𝑟

𝑇 v-speed (not a vector), r-radius of path, T-time for one revolution

𝑎 =𝑣2

𝑟=

4𝜋2𝑟

𝑇2

Also used is frequency which is the reciprocal of the time for one revolution.

Note. There is no such thing as the Centrifugal force. Anyone who disagrees should be shot

and be thrown off a cliff. There is only a centripetal force which is whatever force that is

inducing circular motion.

Questions

1. A Rock is thrown into the air directly upwards at 55m/s by a person standing at the

top of a cliff, and hits the valley after 10s.

i) How long does it take for the rock to reach its maximum height and how high

is that?

ii) How long does it take for the rock to return to its initial height?

iii) How high is the cliff above the Valley

iv) What is the work done moving the rock from the maximum height to the

valley?

2. Two Rocks collide in an elastic collision. Each rock has a mass of 50g and they collide

at right angles to each other with one rock having a speed of 15m/s and the other 35

m/s. Find the resulting speeds of the rocks.

3. An explosion occurs and two pieces are created. One which has a mass of 6kg is

found to be moving at 65 m/s. The other piece is recovered and found to have a

mass of 50g. What speed must it have been moving at and in what direction?