# BMAT Physics Notes: Mechanics

Advice & Insight From BMAT Specialists

## â€‹Scalars & Vectors

AÂ scalarÂ is a quantity that hasÂ magnitudeÂ Â only.
â€‹
AÂ vectorÂ is a quantity that has bothÂ magnitudeÂ andÂ direction.
Â
DISPLACEMENT:Â This is theÂ distanceÂ travelled in a particularÂ direction.

SPEED:Â This is the rate of change ofÂ distanceÂ with respect to time.
It is aÂ scalarÂ quantity.
â€‹
Â

VELOCITY:Â This is the rate of change ofÂ displacementÂ with respect to time.
It is aÂ vectorÂ quantityÂ
â€‹
Â

â€‹ACCELERATION:Â This is the rate of change ofÂ velocityÂ with respect to time.
It is aÂ vectorÂ quantity.
â€‹

## â€‹MOTION GRAPHS

Displacement/distance (d) – time(t) graph.
Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â
In Diagram 1 Â distance doesÂ notÂ increase/decrease from origin.
A â€˜horizontal lineâ€™ means object atÂ rest.

In Diagram 2 theÂ gradientÂ representsÂ  Â Â displacement / distance Â which is theÂ
Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  timeÂ  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â Â
definition of velocity or speed.Â
â€‹
HenceÂ Â  Â Gradient = velocity or speedÂ
Since graphs A and B are bothÂ straight linesÂ then both graphs have constant gradients and henceÂ constantÂ (or uniform)Â velocity/speed.
This means the velocity of A > velocity of B

In Diagram 3,Â  the gradient isÂ increasing.Â This means the speed (or velocity) is increasing or the object is acceleratingÂ Â Â Â Â Â Â Â

In Diagram 4, the gradient isÂ decreasing.
This means the speed (or velocity) is decreasing orÂ decelerating

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Velocity â€“ time graphsÂ Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

In Diagram 1, the velocity does not increase or decrease.
A â€˜horizontal lineâ€™ meansÂ constant velocity.

In Diagram 2 the gradient representsÂ  Â change in velocityÂ  Â which is theÂ
Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â time
definitionÂ ofÂ Â acceleration.
â€‹

HenceÂ Â Â Â Gradient = acceleration
Since the graphs of Â A and B are straight lines then they Â both have constant gradients and henceÂ constantÂ (or uniform) accelerations.
Acceleration of A > acceleration of B

In Diagram 3.

OA ——- constant acceleration.
AB ——- constant velocity
BC ——- constant deceleration

NOTE ALSO:
AREAÂ under graph =Â TOTALÂ DISPLACEMENT/DISTANCE

## EQUATIONS OF MOTION

Suppose a truck is moving as shown below.

â€‹The truck accelerates so that its velocity changes as shown.
Â
u =Â  initial velocityÂ Â Â Â Â  v = final velocityÂ Â Â Â Â Â Â Â Â  a = accelerationÂ Â Â Â Â Â  s =Â  displacementÂ  Â Â Â Â Â t = timeÂ
Â
It can be shown that:
Â

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## ENERGY

Work

Whenever aÂ forceÂ is applied to an object and it moves then we say thatÂ workÂ is done by the person or machine that supplies the force.

The definition of work:
â€‹
Work = Force x distance moved ( in the direction of the force).

Work and Energy
Â
WheneverÂ workÂ is done by a force thenÂ energyÂ is transferred. In fact, we measure the amount of energy given to an object by calculating how muchÂ workÂ is done on it.
Â
In the example of the box being pulled along the ground we say that the person or machine that is pulling the boxÂ transfers energyÂ to the box becauseÂ workÂ is being done on it.
Â
Since the effect of the force on the box is to make itÂ moveÂ we say it gainsÂ kinetic energy (K.E.)
Â
We sayÂ anyÂ moving object has kinetic energy. In the case of the box being pulled along the ground we say the gain in KE of the box is equal to the energy transferred by the person or machine pulling the box. No energy is lost! This is an example of theÂ conservation of energy.
Â
Conservation of energy:Â Energy cannot be created or destroyed but only can be changed from one form into another.
â€‹
Definition of kinetic energy:Â The kinetic energy of a body is theÂ work doneÂ in bringing the bodyÂ toÂ restÂ (or the work done in accelerating the bodyÂ from restÂ to its present speed).

â€‹Kinetic Energy

Definition of kinetic energy:
The kinetic energy of a body is theÂ work doneÂ in bringing the bodyÂ toÂ restÂ (or the work done in accelerating the bodyÂ from restÂ to its present speed).

â€‹Conservation of energy: RememberÂ Energy cannot be created or destroyed but only can be changed from one form into another.

Equation for Gravitational Potential Energy (P.E.)

Suppose a box of mass m is to be lifted from the floor on to a table of height h.
TheÂ weightÂ of the box isÂ mgÂ (see later notes).
If the box is to lifted at aÂ constant speedÂ upwards by a force F,Â  Â then F = mg
TheÂ work doneÂ by the force on the boxÂ  = force x distance = FhÂ Â  where h is theÂ verticalÂ displacement/distance Â of the box.
â€‹
Â work done on box = mgh = gain of (gravitational) potential energy.
Â
HenceÂ Â Â Â Â PE = mgh
Â
By applying theÂ conservation of energyÂ we can state:Â Â loss in PE = gain in KE
Â
We can also state:Â Â Â Â Initial PEÂ  =Â  Final KE

## Power and Efficiency

Power is the rate of doing work or rate of energy transfer

Hence,Â  Â  Power = work (or energy transferred)/timeÂ  Â  Â
Â

Efficiency
Â
Every machine must use some fuel (energy) to operate. HoweverÂ notÂ all the energy supplied to a machine is used for itsÂ usefulÂ purpose. Some energy isÂ wastedÂ usually in the form ofÂ heat.

Efficiency is usually expressed as a percentage:
Â
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Efficiency =Â  Â useful outputÂ Â  x 100 (%)
Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â Â total input

## Forces & Newtonâ€™s laws of Motion

Newtonâ€™s 1st Law of motion

Newtonâ€™s 1st Law of motion:Â An object will remain at rest or in a state of uniform motion (in a straight line) unless acted upon by a resultant external force.
So,
• If an object is atÂ rest,Â then theÂ resultantÂ force acting on it isÂ ZERO.
• If an object moves with aÂ constant (or uniform) velocity (Â orÂ constant speed in a straight line), then theÂ resultantÂ force acting on it is alsoÂ ZERO.

â€‹Newtonâ€™s 2nd Law of Motion
Â
What happens if thereÂ isÂ aÂ resultantÂ force acting on an object?
Newton was able to show that the object willÂ accelerateÂ (orÂ decelerate).

Newtonâ€™s 2nd law:Â  ForceÂ  =Â  mass x accelerationÂ Â  orÂ  F = ma

UNITS:Â  Mass(m) in kg,Â  acceleration (a)Â in ms-2, force (F) inÂ newtons (N).

## â€‹The WEIGHT of an object

The weight of an object is theÂ forceÂ with which the Earth pulls the object.
We can use the 2nd law F = ma to calculate the force on an object and hence itsÂ weight.

â€‹F = maÂ  but a = gÂ  so, F = mg and F is the â€˜weightâ€™ (w)Â Â  HenceÂ Â w = mg
So, an object of mass 1kg weighs 10 N, Â a mass of 2 kg weighs 20 N

## â€‹Alternative meaning for (g)

Since F = mg,Â  it follows thatÂ  g = F/m.Â  F is measured in newtons (N), m is measured in kg.

â€‹When used in this way g is then called Â the gravitational field strength.Â Â It shows how strong the force of gravity is at a particular place.
The force of gravity on the Moon isÂ lessÂ than that on the Earth. It is about 1/6 of the Earthâ€™s.
This means:

Â

â€‹Hence there areÂ twoÂ meanings forÂ g:

• Â The acceleration due to gravity, and
• Â The gravitational field strength.

## Effect of air resistance

Suppose an object, of mass m, is falling under gravity throughÂ air.Â TheÂ weightÂ (Y) of the object acts vertically downwards and isÂ constantÂ at all times. Also Y = mg
Â
However, there is now aÂ  resistive force (X) due toÂ Â frictionÂ between the object and the air. This â€˜air resistanceâ€™ (or â€˜dragâ€™ force) acts in theÂ oppositeÂ direction to the direction of motion, i.e. upwards in this case.

The air resistance isÂ notÂ constant butÂ increasesÂ as the speed of the object increases.

At the start of the fall because the speed isÂ small Â the air resistance is alsoÂ small.Â So, X < Y and there is a resultantÂ downwardÂ force and so the objectÂ accelerates.
Â
As the speed of the object increases, X increases so theÂ resultant downwardÂ force (= Y – X)Â decreasesÂ and so the accelerationÂ decreases.
Â
Eventually, if the object falls fast enough (and it may hit the ground before this happens), the air resistance/drag Â willÂ equalÂ the weight of the object i.e.Â X = YÂ and so theÂ  resultant force acting on the object will beÂ zero.Â The acceleration will then also beÂ zeroÂ and the object will continue to fall at aÂ constant speed.Â This speed is usually called theÂ terminal velocity.

A typical speed-time graph for a falling object is shown.
Â
Factors that affect the value of air resistance are

• speed of object,
• shape and area of object,
• density of air
â€‹Example 1
Â
Â
â€‹The diagram shows a parachutist falling from an aeroplane and the velocity-time graph of the motion.
Explain, in terms of the forces acting the shape of the graph.
Â
YÂ is theÂ weightÂ of the person and isÂ constantÂ throughout the motion.
XÂ is theÂ air resistance/dragÂ and canÂ changeÂ during the motion.
Â
AB:Â  Initially, Y > X. As the person acceleratesÂ  XÂ increases.Â This causes theÂ resultantÂ force ( Y â€“ X) toÂ decrease.Â Hence the acceleration decreases (from F = ma). This causes theÂ gradientÂ (which is equal to theÂ acceleration) to decrease.
Â
BC:Â Â AtÂ B, the air resistanceÂ XÂ equalsÂ the weightÂ YÂ so the resultant force is zero. Hence the person falls at aÂ constant speedÂ (theÂ terminalÂ velocity).
Â
CD:Â AtÂ CÂ the parachute is opened. This greatly increases the air resistanceÂ XÂ because of the largeÂ AREAÂ of surface. Now Y < X and so there is a resultantÂ upwardÂ force. This causes the person toÂ decelerate.Â So the speed decreases. As the speed decreases the value ofÂ XÂ decreases.
Â
DE:Â AtÂ D, the value ofÂ XÂ has decreased to a value such that it now again equalsÂ Y.Â The resultant force is again zero soÂ  that the personÂ  moves at a lower constant speed.
##### Newtonâ€™s 3rd Law of motion

Newton realised that forces always exist inÂ pairs.
That is to say that whenÂ oneÂ body experiences a force then an equal but opposite force will be created onÂ anotherÂ body.
This is known asÂ Newtonâ€™s 3rd law of motionÂ which is more formally given below.

If body A exerts a force on body B then body B exerts an equal and opposite force of the same type on body A.

## MOMENTUM (p)

Â Momentum is defined by:Â Â Â Â Â momentum = massÂ xÂ velocityÂ  (p = mv)
It is aÂ vectorÂ quantity.Â Â

Conservation of momentum:

In any interaction between objects the total momentum in any direction is constant provided no external force acts.

Newtonâ€™s 2nd law and Momentum

We have seen that the 2nd law is given by F = ma.
However, Newton explained the law in terms of momentum.

â€‹Force = rate of change of momentumÂ Â  orÂ Â  Force =Â Â Â change in momentumÂ
Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â Â Â time

The effect of forces on Materials

Suppose a mass is connected to a spring as shown.
The spring willÂ stretch.
If the mass is increased then the stretch will increase.

ForÂ someÂ materials if a graph of theÂ force (F)Â applied (theÂ load) is plotted against theÂ extension (x)Â then aÂ straight lineÂ passing through the origin is obtained.

This is known asÂ Hookeâ€™s lawÂ and is stated below.

Hookeâ€™s law:Â  The extension is directly proportional to the force applied.

This is given as an equation:Â  Â  Â F = k x

Where k is a constant called theÂ Â force constant.

What feature of the graph gives k ?Â Â Â Â Â  __________________________

Stored Energy in a stretched material

When a material experiences a stretching forceÂ workÂ is done on the material by the applied force. This force causes the atoms of the material to beÂ displacedÂ from their original positions and so they gainÂ internal potentialÂ  (orÂ elastic) energy.

It can be shown that the stored energy is given by the following equation:

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Â Â Â Â Â STORED ENERGYÂ  =Â  Â½ F xÂ Â Â Â Â

(Note:Â FÂ must be inÂ newtonsÂ andÂ xÂ inÂ metresÂ for this equation to give energy inÂ J).

However,Â  since F = kx, then if we substitute for F in the above equation we obtain the equation:

Force-extension Graph
Â
We have seen that for some materials a graph of force against extension is aÂ straight line (OP).
In this region the force is directly proportional to the extension (Hookeâ€™s law).
If the material is stretched beyond P it is no longer a straight line.
For this reason pointÂ PÂ is called theÂ limit of proportionality.
â€‹
â€‹If the materialÂ  returns to its original shape and size when the force is removed the stretching is calledÂ elastic.Â

This may occur up to a pointÂ EÂ called theÂ elastic limit.
If the material is stretched beyond E it willÂ notÂ return to its original size after the force is removed.
This region (ET)Â  is called theÂ inelasticÂ region.
The region (OPE) is called theÂ elasticÂ region.

#### BMAT Physics Notes: Mechanics

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