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BMAT Physics Notes: Mechanics

Advice & Insight From BMAT Specialists

​Scalars & Vectors

scalar is a quantity that has magnitude  only.

vector is a quantity that has both magnitude and direction.
 
DISPLACEMENT: This is the distance travelled in a particular direction.

SPEED: This is the rate of change of distance with respect to time.
It is a scalar quantity.
Picture
 

VELOCITY: This is the rate of change of displacement with respect to time.
It is a vector quantity 
Picture
 

​ACCELERATION: This is the rate of change of velocity with respect to time.
It is a vector quantity.
Picture

​MOTION GRAPHS

Displacement/distance (d) – time(t) graph.
                                                                                           
In Diagram 1  distance does not increase/decrease from origin.
A ‘horizontal line’ means object at rest.

In Diagram 2 the gradient represents    displacement / distance  which is the 
                                                                              time                      
definition of velocity or speed. 

Hence    Gradient = velocity or speed 
Since graphs A and B are both straight lines then both graphs have constant gradients and hence constant (or uniform) velocity/speed.
Gradient of A > gradient of B
This means the velocity of A > velocity of B

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

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

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Velocity – time graphs                                            

In Diagram 1, the velocity does not increase or decrease.
A ‘horizontal line’ means constant velocity.

In Diagram 2 the gradient represents   change in velocity   which is the 
                                                                       time
definition of  acceleration.

Hence    Gradient = acceleration
Since the graphs of  A and B are straight lines then they  both have constant gradients and hence constant (or uniform) accelerations.
Gradient of A > gradient of B
Acceleration of A > acceleration of B

In Diagram 3.

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

NOTE ALSO:
AREA under graph = TOTAL DISPLACEMENT/DISTANCE

EQUATIONS OF MOTION

Suppose a truck is moving as shown below.

​The truck accelerates so that its velocity changes as shown.
 
u =  initial velocity      v = final velocity          a = acceleration       s =  displacement       t = time 
 
It can be shown that:
 
Picture

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ENERGY

Work

Whenever a force is applied to an object and it moves then we say that work is done by the person or machine that supplies the force.

The definition of work:

Work = Force x distance moved ( in the direction of the force).

Work and Energy
 
Whenever work is done by a force then energy is transferred. In fact, we measure the amount of energy given to an object by calculating how much work is done on it.
 
In the example of the box being pulled along the ground we say that the person or machine that is pulling the box transfers energy to the box because work is being done on it.
 
Since the effect of the force on the box is to make it move we say it gains kinetic energy (K.E.)
 
We say any 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 the conservation of energy.
 
Conservation of energy: Energy cannot be created or destroyed but only can be changed from one form into another.

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

​Kinetic Energy


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

Picture

​Conservation of energy: Remember 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.
The weight of the box is mg (see later notes).
If the box is to lifted at a constant speed upwards by a force F,   then F = mg
The work done by the force on the box  = force x distance = Fh   where h is the vertical displacement/distance  of the box.

 work done on box = mgh = gain of (gravitational) potential energy.
 
Hence     PE = mgh
 
By applying the conservation of energy we can state:  loss in PE = gain in KE
 
We can also state:    Initial PE  =  Final KE

Power and Efficiency

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

Hence,    Power = work (or energy transferred)/time     
 
Picture

Efficiency
 
Every machine must use some fuel (energy) to operate. However not all the energy supplied to a machine is used for its useful purpose. Some energy is wasted usually in the form of heat.

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

Forces & Newton’s laws of Motion

Newton’s 1st Law of motion

Newton’s 1st Law of motion: 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 at rest, then the resultant force acting on it is ZERO.
  • If an object moves with a constant (or uniform) velocity ( or constant speed in a straight line), then the resultant force acting on it is also ZERO.

​Newton’s 2nd Law of Motion
 
What happens if there is resultant force acting on an object?
Newton was able to show that the object will accelerate (or decelerate).

Newton’s 2nd law:  Force  =  mass x acceleration   or  F = ma

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

​The WEIGHT of an object

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

Picture

​F = ma  but a = g  so, F = mg and F is the ‘weight’ (w)   Hence  w = mg
So, an object of mass 1kg weighs 10 N,  a mass of 2 kg weighs 20 N

​Alternative meaning for (g)

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

Picture

​When used in this way g is then called  the gravitational field strength.  It shows how strong the force of gravity is at a particular place.
The force of gravity on the Moon is less than that on the Earth. It is about 1/6 of the Earth’s.
This means:

Picture

 

​Hence there are two meanings for g:

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

Effect of air resistance

Suppose an object, of mass m, is falling under gravity through air. The weight (Y) of the object acts vertically downwards and is constant at all times. Also Y = mg
 
However, there is now a  resistive force (X) due to  friction between the object and the air. This ‘air resistance’ (or ‘drag’ force) acts in the opposite direction to the direction of motion, i.e. upwards in this case.

The air resistance is not constant but increases as the speed of the object increases.

At the start of the fall because the speed is small  the air resistance is also small. So, X < Y and there is a resultant downward force and so the object accelerates.
 
As the speed of the object increases, X increases so the resultant downward force (= Y – X) decreases and so the acceleration decreases.
 
Eventually, if the object falls fast enough (and it may hit the ground before this happens), the air resistance/drag  will equal the weight of the object i.e. X = Y and so the  resultant force acting on the object will be zeroThe acceleration will then also be zero and the object will continue to fall at a constant speedThis speed is usually called the terminal velocity.

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

  • speed of object,
  • shape and area of object,
  • density of air
​Example 1
 
Picture
 
​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.
 
is the weight of the person and is constant throughout the motion.
is the air resistance/drag and can change during the motion.
 
AB:  Initially, Y > X. As the person accelerates  X increases. This causes the resultant force ( Y – X) to decrease. Hence the acceleration decreases (from F = ma). This causes the gradient (which is equal to the acceleration) to decrease.
 
BC:  At B, the air resistance X equals the weight Y so the resultant force is zero. Hence the person falls at a constant speed (the terminal velocity).
 
CD: At C the parachute is opened. This greatly increases the air resistance because of the large AREA of surface. Now Y < X and so there is a resultant upward force. This causes the person to decelerate. So the speed decreases. As the speed decreases the value of decreases.
 
DE: At D, the value of has decreased to a value such that it now again equals Y. The resultant force is again zero so  that the person  moves at a lower constant speed.
Newton’s 3rd Law of motion

Newton realised that forces always exist in pairs.
That is to say that when one body experiences a force then an equal but opposite force will be created on another body.
This is known as Newton’s 3rd law of motion 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)

 Momentum is defined by:     momentum = mass x velocity  (p = mv)
It is a vector quantity.  

Conservation of momentum:

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

Newton’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.

​Force = rate of change of momentum   or   Force =   change in momentum 
                                                                                                     time

The effect of forces on Materials

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

For some materials if a graph of the force (F) applied (the load) is plotted against the extension (x) then a straight line passing through the origin is obtained.

This is known as Hooke’s law and is stated below.

Hooke’s law:  The extension is directly proportional to the force applied.

This is given as an equation:     F = k x

Where k is a constant called the  force constant.

Picture

What feature of the graph gives k ?      __________________________

Stored Energy in a stretched material

When a material experiences a stretching force work is done on the material by the applied force. This force causes the atoms of the material to be displaced from their original positions and so they gain internal potential  (or elastic) energy.

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

                         STORED ENERGY  =  ½ F x     

(Note: F must be in newtons and x in metres for this equation to give energy in J).

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

Force-extension Graph
 
We have seen that for some materials a graph of force against extension is a straight line (OP).
In this region the force is directly proportional to the extension (Hooke’s law).
If the material is stretched beyond P it is no longer a straight line.
For this reason point P is called the limit of proportionality.
​If the material  returns to its original shape and size when the force is removed the stretching is called elastic. 

This may occur up to a point E called the elastic limit.
If the material is stretched beyond E it will not return to its original size after the force is removed.
This region (ET)  is called the inelastic region.
The region (OPE) is called the elastic region.

BMAT Physics Notes: Mechanics

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