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PREPARED NOTES

# INTRODUCTION:

Increased access to transport is a feature of today’s society. Most people access some form of transport for travel to and from school or work and for leisure outings at weekends or on holidays. When describing journeys that they may have taken in buses or trains, they usually do so in terms of time or their starting point and their destination. When describing trips they may have taken in planes or cars, they normally use the time it takes, distance covered or the speed of the vehicle as their reference points. While distance, time and speed are fundamental to the understanding of kinematics and dynamics, very few people consider a trip in terms of energy, force or the momentum associated with the vehicle, even at low or moderate speeds.

The faster a vehicle is travelling, the further it will go before it is able to stop. Major damage can be done to other vehicles and to the human body in collisions, even at low speeds. This is because during a collision some or all of the vehicle’s kinetic energy is dissipated through the vehicle and the object with which it collides. Further, the materials from which vehicles are constructed do not deform or bend as easily as the human body. Technological advances and systematic study of vehicle crashes have increased understanding of the interactions involved, the potential resultant damage and possible ways of reducing the effects of collisions. There are many safety devices now installed in or on vehicles, including seat belts and air bags. Modern road design takes into account ways in which vehicles can be forced to reduce their speed.

The branch of Physics that deals with phenomena such as those mentioned above is called mechanics.

Note: Some internet browsers (eg Firefox) do not accurately display text symbols such as Greek letters used to represent quantities in Physics.  For example, capital delta is displayed as D in Firefox.  So, if you see below something like Ds/Dt for speed or velocity, it is meant to appear as delta s/delta t.  This is just something to be aware of in case you do come across such issues.  The square root sign is another one not displayed properly by some browsers.  Any symbols used in equations produced by equation editors will of course display properly.

# MECHANICS:

The branch of Physics that is concerned with the motion and equilibrium of bodies in a particular frame of reference is called “mechanics”.  Mechanics can be divided into three branches: (i) Statics – which deals with bodies at rest relative to some given frame of reference, with the forces between them and with the equilibrium of the system; (ii) Kinematics - the description of the motion of bodies without reference to mass or force; and (iii) Dynamics – which deals with forces that change or produce the motions of bodies.

Some common terms used in the study of mechanics (and indeed many other branches of Physics) are: scalars, vectors and SI Units.

1.      Scalars – A scalar is a physical quantity defined in terms of magnitude (size) only - eg temperature, mass, volume, density, distance.

2.      Vectors – A vector is a physical quantity defined in terms of both magnitude and direction - eg force, velocity, acceleration, electric field strength.   Diagramatically we can represent a vector by a straight line with an arrow on one end.  The length of the line represents the magnitude of the vector quantity and the direction in which the arrow is pointing represents the direction of the vector quantity.  We will say much more about vectors later in this topic.

3.      Systeme International (SI) Units – The internationally agreed system of units.  There are seven fundamental units.  The three that we will use in this topic are the metre (length), the kilogram (mass) and the second (time).  Various prefixes are used to help express the size of quantities eg a nanometre (1 nm) = 10-9 of a metre, a gigametre (1 Gm) = 109 metres.  See Appendix C, p.243 of Excel Preliminary Physics by Warren for a list of some common SI units and prefixes.  Most texts will contain such information.

Since describing the features of something is usually a lot simpler than explaining how or why it works, we will start with a look at kinematics.

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# MOTION:

The following terms are commonly used to describe motion.

1.      Displacement – is the distance of a body from a given point in a given direction.  It is a vector quantity.  The SI unit of displacement is the metre (m).

2.      Speed – The speed of a body is the rate at which it is covering distance.  It is a scalar quantity.  The SI units are m/s, which can also be written as ms-1.

where vav = average speed, d = total distance travelled and t = total time taken to travel distance d.

3.      Velocity – The velocity of a body is its speed in a given direction.  In other words, velocity is the rate of change of displacement with time.  It is a vector quantity with the same SI units as speed.

where vav = average velocity, Dr = change in displacement and Dt = change in time taken to achieve that change in displacement.

Another way to express average velocity is as the average of the initial and final velocities.

where vav = average velocity, u = initial velocity of the body and v = final velocity of the body.  Note that this equation applies ONLY when the velocity of the body is increasing or decreasing at a constant rate.

4.      Acceleration – The acceleration of a body is the rate of change of the velocity of the body with time.  It is a vector quantity, with units of (metres/second)/second, written as ms-2.

where aav = average acceleration, Dv = change in velocity of the body and Dt = change in time over which the change in velocity took place.  Alternatively, we may write:

where aav = average acceleration, v = final velocity of the body, u = initial velocity of the body, and t = time over which the change in velocity took place.

Note that a body accelerates when:

a.      It speeds up;

b.      It slows down;

c.       It changes direction.

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# MOTION GRAPHS:

Often the most effective way to describe the motion of a body is to graph it.  Note that in this section the variable “s” will be used to represent displacement instead of “r”.  You will find in Physics that these two variables are both in common use to denote displacement.

## Displacement-Time Graphs

These may be used to gain information about the displacement of an object at various times or about the velocity of the object at various times.

Clearly, the gradient (slope) of a displacement-time graph gives the velocity.

Note that a positive gradient implies a positive velocity and a negative gradient implies a negative velocity.

For a curved displacement-time graph, the gradient of the tangent to the curve at a particular point equals the gradient of the curve at that point, which in turn equals the velocity of the object at that particular time.  Such a velocity, that is, the velocity at a particular instant in time, is called the instantaneous velocity.

An example of an instrument that measures instantaneous velocity is the speedometer in a car.  In older cars the speedometer was linked mechanically to the transmission. These days, however, a device located in the transmission produces a series of electrical pulses whose frequency varies in proportion to the vehicle's speed.  The electrical pulses are sent to a calibrated device that translates the pulses into the speed of the car. This information is sent to a device that displays the vehicle's speed to the driver in the form of a deflected speedometer needle or a digital readout.

Note that a straight line displacement-time graph implies that velocity is constant.  A curved line displacement-time graph implies that velocity is changing with time (ie the object is accelerating).

## Velocity-Time Graphs

These may be used to gain information about the displacement, velocity and acceleration of an object at various times.

The gradient is clearly the acceleration of the object.

Note that a positive gradient implies a positive acceleration and a negative gradient implies a negative acceleration.

Also, the area under the graph,

in the case above, has units of: seconds x metres per second = metres.  Thus, the area under a velocity-time graph is equal to the displacement travelled by the object in the time Dt.

Note that a horizontal straight line velocity-time graph implies that acceleration is zero – ie velocity remains constant.

A non-horizontal, straight line velocity-time graph implies that acceleration is constant and non-zero.

A curved line velocity-time graph implies that acceleration is varying.

## Acceleration-Time Graphs

These may be used to gain information about the velocity and acceleration of an object at various times.

The area under an acceleration-time graph gives the change in velocity of an object during the time interval Dt.  Check the units of the area: (ms-2 x s = ms-1).

A horizontal straight line acceleration-time graph implies that velocity is varying at a constant rate (ie velocity is increasing or decreasing by the same amount each second).  That is, acceleration is constant.

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# RELATIVE VELOCITY:

Often it is necessary to compare the velocity of one object to that of another.  For instance, two racing car drivers, A and B, may be travelling north at 150 km/h and 160 km/h respectively.  We could say that the velocity of car B relative to car A is 10 km/h north.  In other words, driver A would see driver B pull away from her with a velocity of 10 km/h north.

Likewise, two jet aircraft, C and D, flying directly at each other in opposite directions (hopefully as part of an aerobatics display) may have velocities of 900 km/h north and 1000 km/h south respectively.  We could say that the velocity of D relative to C is 1900 km/h south.  In other words, jet C will observe jet D flying towards it at a speed of 1900 km/h.

Clearly, when the objects are travelling in the same direction, the velocity of one relative to the other is the difference between their speeds, taking due care to state the appropriate direction.  When the objects are travelling in opposite directions the velocity of one relative to the other is the sum of their speeds, again taking due care to state the correct direction.  There is a vector equation which can be used to calculate the relative velocities of objects, even when the objects travel at various angles to one another but this equation is outside the scope of the present syllabus (for some unfathomable reason).

Check out the "Flight of Fantasy" Brain Teaser.

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# FORCE:

## What is Force?

In simple terms a force can be defined as a push or a pull.  We experience examples of forces every day.  If we push a stationary lawn mower (with enough force) it begins to move – that is, it accelerates and its velocity increases.  If we push on the brakes of a moving bicycle, it slows down – that is it decelerates (or undergoes negative acceleration).  If we apply sufficient force to an aluminum can, by squeezing it with our hand, we can change the shape of the can.

So a force can cause a change in the state of motion of an object or a change in the shape of an object.  In fact, all accelerations (and decelerations) are caused by forces.

## Does every force cause acceleration?

Again, from our everyday experience, we know the answer to this question is “no”. If a person pushes on the brick wall of a house, the house does not accelerate.  Sometimes when we want to push or pull an object from one place to another we find that no matter how hard we push or pull, we just cannot move (accelerate) the object.

In previous Science courses, a qualitative relationship was established between force, mass and acceleration.  In the Preliminary Physics Course we need to establish a quantitative relationship between these three quantities.

## What is the relationship between force and acceleration?

We could perform an experiment to determine the relationship between the size of a force applied to an object at rest on a laboratory bench and the change in velocity experienced by the object over a set period of time (ie the acceleration).  Such an experiment would produce results as shown below.

The graph above shows that:

¨      The change in velocity does not happen instantaneously. A certain amount of force is required before the object begins to accelerate.  This makes sense, since the force of friction between the bench and the object must be overcome before the object can move.  So, we can say that a net external force is required in order to change the velocity of an object.

¨      The acceleration produced is directly proportional to the force applied.  If we repeated the experiment on a frictionless surface (eg using a dry-ice puck on a very smooth, polished table top) the straight-line graph would even pass through the origin.

## What is the relationship between acceleration and mass?

We could measure the accelerations produced when the same sized force is applied to different objects.  Such an experiment would produce results like those below.

The graph above suggests that there is an inverse relationship between acceleration and mass.  A plot of acceleration versus the reciprocal of mass, using the same data, would produce a graph similar to that below.

This graph clearly shows that:

¨      The acceleration produced by a given force is inversely proportional to the mass of the object.

From experiments such as those above, we can say that:

and

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FOR SOME FUN: View this video of a person walking on eggs.

# NEWTON’S LAWS OF MOTION

## Newton’s First and Second Laws:

By combining the results above and defining the units of force appropriately, we can write that:

.

This can be taken as a statement of Newton’s Second Law.  The SI Unit of force is the newton (N), defined so that 1N = 1kgms-2.

Note that in the above equation, F is the vector sum of all the forces acting on the object, m is the mass of the object and a is its vector acceleration.  To remind us of that fact we will write:

.

Note that if the resultant force on the object is zero, there is no acceleration.  Therefore, in the absence of a resultant force, an object’s velocity will remain unchanged.  In other words, an object at rest will remain at rest, and an object in motion will remain in motion with uniform velocity, unless acted upon by a net external force.  This is a statement of Newton’s First Law, which in fact is contained in the Second Law as a special case (for SF = 0).

## Newton’s Third Law:

Forces acting on a body originate in other bodies that make up its environment.  Any single force is only one aspect of a mutual interaction between two bodies.  We find by experiment that when one body exerts a force on a second body, the second body always exerts a force on the first.  Furthermore, we find that these forces are equal in size but opposite in direction.  A single, isolated force is therefore an impossibility.

If one of the two forces involved in the interaction between two bodies is called an action force, the other is called the reaction force.  Either force may be called the action and the other the reaction.  Cause and effect is not implied here, but a mutual simultaneous interaction is implied.

This property of forces was first stated by Newton in his Third Law: “To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.”

In other words, if body A exerts a force on body B, body B exerts an equal but oppositely directed force on body A; and furthermore the forces lie along the line joining the bodies.  Notice that the action and reaction forces, which always occur in pairs, act on different bodies.  If they were to act on the same body, we could never have accelerated motion because the resultant force on every body would be zero.

Consider the following examples:

1.      Imagine a boy kicking open a door.  The force exerted by the boy B on the door D accelerates the door (it flies open); at the same time, the door D exerts an equal but opposite force on the boy, which decelerates the boy (his foot loses forward velocity).  The force of the boy on the door and the force of the door on the boy is an action-reaction pair of forces.

2.      When you walk, you apply a force backwards on the earth.  Likewise, the earth applies a force to you of equal magnitude but in the opposite direction.  So, you move forwards.

The force of the person on the earth and the force of the earth on the person is an action-reaction pair of forces.

3.      Consider a body at rest on a horizontal table:

Each of the pairs of forces above is an action-reaction pair of forces.

## Definitions of Mass and Weight:

The mass of an object is a measure of the amount of matter contained in the object.  Mass is a scalar quantity.

The weight of an object is the force due to gravity acting on the object.  Weight is a vector quantity.

The weight, W, of an object is given by Newton’s 2nd Law as:

where m is the mass of the object and g is the acceleration due to gravity (9.8 ms-2 close to the earth’s surface).

## USEFULNESS OF VECTOR DIAGRAMS:

Many of the quantities with which we deal in Physics are vectors.  Sometimes we need to add a number of vectors together.  For instance, we may be trying to calculate the total or resultant force acting on a car when several forces act on the car simultaneously – the wind, friction, gravity and the force supplied by the engine.  Sometimes we need to subtract two vectors.  For instance, we may be trying to calculate the change in velocity of a car as it goes around a bend in the road.  The change in velocity of the car equals the final velocity of the car minus initial velocity of the car.

When the need arises to add or subtract vector quantities, this proves to be easy only when the vector quantities act along the same straight line.  If the vectors act at an angle to each other we really need to draw a vector diagram to assist in solving the problem.

Vector analysis is an extremely important aspect of Physics and there are several different methods available to add, subtract and even multiply vectors.  Unfortunately, the current Syllabus requires that you have only a very basic understanding of vector analysis.  So, we will examine a single, simple but very useful method of adding and subtracting vectors.

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The method we will use is called the “Vector Polygon” method.  To find the sum of a number of vectors draw each vector in the sum, one at a time, in the appropriate direction, placing the tail of the second vector so that it just touches the head of the first.  Continue in this fashion until all of the vectors in the sum have been included in the diagram.  Note that it does not matter which vector you start with.

The vector that closes the vector polygon in the same sense as the component vectors is called the equilibrant.  It is the vector which when drawn into the diagram gets you back to where you started.  The vector that closes the vector polygon in the opposite sense to the component vectors is called the resultant.  The resultant is the answer to the sum of all the vectors.  Its size can be calculated mathematically or measured using a ruler if the vector polygon has been drawn to scale.  The direction of the resultant can be calculated mathematically or can be measured using a protractor if the vector polygon has been drawn to scale.  Either way, the direction of the resultant must be stated in an unambiguous way.

Sometimes in Physics our vector additions only involve two vectors at a time.  In this case, the polygon formed is a triangle, making the mathematical calculation of the magnitude and direction of the resultant quite straight forward.  So, keep your wits about you and bring your knowledge of triangle geometry and trigonometry into the Physics classroom.

EXAMPLE 1: A fighter pilot flies her F-14D Tomcat jet with a true airspeed of 400 km/h North.  A crosswind from the East blows at 300 km/h relative to the ground.  Calculate the jet’s resultant velocity relative to the ground.

Note: For aircraft, the true airspeed (TAS) is the actual speed of the aircraft through the air (the speed of the aircraft relative to the air).  The wind speed is usually measured relative to the ground. Groundspeed is the speed of the aircraft relative to the ground.  The groundspeed of the aircraft is the vector sum of the true airspeed and the wind speed.

Obviously, a vector diagram would be very useful in solving Example Problem 1.  See the diagram below.

By Pythagoras’ Theorem, the magnitude of the resultant velocity of the jet is:

and the direction can be found using basic trigonometry as follows:

So, the velocity of the jet relative to the ground is 500 km/h N36.9oW.

Note that if the angle between the two vectors being added together is other than 90o, Cosine Rule and Sine Rule can be used to solve the problem mathematically.  Note also the use of the compass in the diagram to establish direction.

EXAMPLE 2: In the previous problem, in which direction should the pilot head and with what airspeed in order to actually fly north at 400 km/h relative to the ground?

Again a vector diagram is useful.  Our intuition tells us that the pilot must fly into the wind.  So, when we draw a diagram that shows all of the information that we know to be true, we obtain the diagram shown below.

Clearly, if the pilot flies N36.9oE with an airspeed of 500 km/h, the wind will bring her back to a heading of due North at a speed of 400 km/h relative to the ground.  Remember also, there is usually more than one way to give the direction.  The direction the pilot should fly in this example could just as correctly be given as E53.1oN or as a True Bearing of 36.9o.

EXAMPLE 3: Four children pull on a small tree stump firmly stuck in the ground.  Looking down on the tree stump from above, the forces applied by the children are as shown below.

Determine the resultant force applied to the tree stump.

To solve this problem mathematically we would need to add two of the vectors together, then add our answer to the third vector and finally add our answer to that addition to the fourth vector.  It is actually far quicker and easier to solve this problem graphically.  To do this we construct a vector polygon, using the rules stated above and simply measure the size and direction of the resultant.  See the diagram below.

The resultant force, R, is found by measurement to be 3.1 N at an angle of 39o clockwise from the direction of the 4.5 N force.

Note that when using a graphical approach, the scale must be clearly stated on the diagram.  Always choose a sensible numerical scale.  Also, choose a scale that will produce a large diagram.  The larger the diagram, the more accurate the answer.  For the example problem above, the scale used was 1 N = 1.5 cm.  This is certainly the smallest scale I would use for this particular problem.  Anything less is too inaccurate.  A scale of 1 N = 2 cm would be preferable.  The smaller scale was used here to fit the diagram neatly onto this page.

Note also, that depending on which printer is used to print these notes, there may be a small discrepancy between the stated scale and the actual scale on the page.

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## VECTOR SUBTRACTION:

If vector A is as shown below:

then vector –A is a vector of the same magnitude as A but opposite direction.

In order to find the difference between two vectors, add the negative of the second vector to the first.

EXAMPLE 4: A Maserati (car) is moving due East at 20 ms-1.  A short time later it is moving due North at 20 ms-1.  Calculate the change in velocity of the Maserati.

To find the change in size of any quantity, you subtract the initial size of the quantity from the final size.  Obviously, with vector quantities you must do a vector subtraction not just an arithmetic one, since vectors possess both size and direction.

Change in velocity = final velocity – initial velocity

This should really be written as:

Change in velocity = final velocity + (– initial velocity)

since that is how we draw the vector diagram.  We literally add the negative of the initial velocity to the final velocity.  Study the vector diagram below to ensure you understand the process of vector subtraction.

Using Pythagoras’ Theorem and basic trigonometry as shown in Example 1 above, we find that the change in velocity of the Maserati is 28.3 ms-1 at an angle of 45o West of North.

Note that even though the car has the same initial and final speeds, because the direction of the car has changed, so too has its velocity.

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## VECTOR COMPONENTS:

Sometimes we are only interested in part of a vector rather than all of it.  For instance, if we push a car that has run out of petrol, we apply a force to the car.  However, if we are not careful some of the force we apply pushes down vertically on the car and the rest of it pushes horizontally on the car.  Obviously, we are trying to maximise the component of the force that is applied horizontally.  The angle at which we apply force to the car will determine how much of our force is applied horizontally.

Any vector may be broken into two component vectors at right angles to each other.  These components are called the rectangular components of the vector.  The rectangular components of a vector add up to the original vector.

Consider, the example we used above.  We may push on the car with a force F at an angle of q to the horizontal, as shown below.

The force F may be broken into its vertical and horizontal components as shown below.

The magnitude of each component can then be calculated using simple trigonometry.  The size of the vertical component of F is Fsinq.  The size of the horizontal component of F, the one that must overcome the force of friction if we are to move the car forward, is Fcosq.

EXAMPLE 5: A block of mass 20 kg is being pulled up an inclined plane by a rope inclined at 30o to the plane’s surface as shown in the diagram below.

The plane is inclined at 45o to the horizontal.  The friction force, F, opposing the block’s motion is 10 N.  Determine the tension, T, in the rope if the net acceleration, a, of the block up the plane is 4 ms-2.

You will observe that we have resolved two vectors in the diagram into rectangular components – the tension, T, in the rope and the weight, W, of the block.  The rectangular components we have chosen are those acting parallel to and perpendicular to the inclined plane.  These components are the most useful ones in a situation like this.

Now the total force acting down the plane is the sum of the friction force, F, and the component of the weight force of the block acting down the plane (W sinq).  So, from the diagram we have:

FD = F + mg sinq (since W = mg)

FD = 10 + 20 x 9.8 x sin 45o

FD = 148.59 N

Total force acting up the plane must be:

FU = ma +  FD

FU = (20 x 4) + 148.59

FU = 228.59 N

Note that the logic we have used to obtain an expression for FU is as follows.  The force up the plane MUST be sufficient to overcome the 148.59 N force down the plane and to provide the required force of 80 N to give the block the correct acceleration up the plane.

Now we are in a position to calculate the tension in the rope.  The total force up the plane FU is actually the component of the tension force parallel to the plane.  This is the part of the tension force that is applied parallel to the plane.

Therefore, we have:

cos 30o =  FU / T

T = FU /cos 30o

T = 228.59 / cos 30o

T = 263.95 N

So, the tension in the rope works out to be 264 N.

You are now ready to try Vector Analysis Worksheet No.1.

You may also like to check out this applet, which helps you to understand vector addition & subtraction.  When you get to the site, click on the "Applet Menu" button at the top left of the page.  The menu items will appear.  Select "Some Math", "Vectors", "Vector Addition".  Read the instructions & run the applet.

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# Tensions In Strings (Extension Topic)

The following section is not essential to the Syllabus.  It is however a very good section to do with students if time permits.  It gives students an opportunity to expand their understanding of both the usefulness of vectors in Physics and how to analyze objects under the influence of gravity in different situations.

Consider a mass, m, supported by a thin, inextensible string of negligible mass.  The two forces acting on the mass are T, the tension in the string acting upwards and mg, the weight force acting downwards on the mass.

The vector equation describing the net force acting on the mass is best studied in three separate cases.

Case 1: The mass is supported by the string and there is no acceleration.  The mass could be at rest or could be moving up or down at constant velocity.  In each case the net force equation is the same.  The tension upwards is exactly balancing the weight force downwards.

Case 2: The mass is accelerating downwards with net acceleration, a.  The tension upwards in the string is not sufficient to fully balance the weight force downwards.

Clearly, with the mass accelerating downwards, the tension, T = m ( g – a ).

Case 3: The mass is accelerating upwards with net acceleration, a.  Here, the tension upwards in the string is doing two jobs.  It is fully balancing the weight force downwards and supplying the required force upwards to accelerate the mass with the net acceleration of a.

Clearly, with the mass accelerating upwards, the tension, T = m ( g + a ).

Many problems in Physics can be solved by applying the knowledge summarized above.  Consider the following two examples.

EXAMPLE 1: Two masses X of 10kg and Y of 24kg are connected by a light inextensible string.

X and Y hang on opposite sides of a frictionless pulley as shown above. Determine:
(a)       the net acceleration of the system of masses
(b)       the magnitude of the tension, T, in the string.

Assume that the acceleration due to gravity is 9.8 ms-2.

SOLUTION: There are two slightly different approaches possible.

Solution 1: The Intuitive Approach

Firstly, determine the net force on the system, the total mass of the system and then obtain the net acceleration from F = ma.  Once the net acceleration of the system is known the tension in the string can be found by realising that the tension upwards on the left side of the pulley must balance the weight force down on the mass & supply sufficient force to accelerate the mass upwards with the net acceleration of the system.

(a) Force of gravity on 10kg mass = 10 x 9.8 = 98N down on left side of pulley.

Force of gravity on 24kg mass = 24 x 9.8 = 235.2N down on right side of pulley.

Thus, the net force, F, applied to the system of two masses by gravity:

F = 235.2 – 98  =  137.2 N down on right side of pulley.

Total mass of system upon which this net force acts = 10 + 24 = 34kg.

Therefore from F = ma, the net acceleration of the system of two masses is:

a = F / m  = 137.2 / 34  =  4.035 ms-2.

Note that as you get used to using this method, it really only takes a couple of lines of working at the most.

(b) Once the acceleration is known the tension can be calculated from either mass.  Let’s use the 10kg mass first.  Clearly, the tension in the string on this side of the pulley must be sufficient to balance the acceleration due to gravity down on the 10kg mass and to accelerate the 10kg mass upwards at 4.035 ms-2.  Therefore,

Tension, T = (10 x 9.8) + (10 x 4.035) = 138.35 N upwards.

OR if we decided to use the 24kg mass instead - the tension in the string on the right side of the pulley must be sufficient to balance the acceleration due to gravity down on the 24kg mass less the 4.035 ms-2 that the mass is being permitted to accelerate downwards already.  Therefore,

Tension, T = (24 x 9.8) – (24 x 4.035) = 138.36 N upwards.

This is the same (to one decimal place) as the answer we obtained using the other mass.  This must be the case.  It does not matter which mass you use to calculate the tension, you must get the same answer in both cases because there is only one string and can therefore be only one tension.

The tension in the string is therefore 138.4 N to one decimal place.  Any discrepancy in the above answers after the first decimal place is simply due to rounding off the (137.2 / 34) calculation for the net acceleration in the first place.

Again I stress that the whole solution (parts a & b) I have demonstrated here would normally take no more than four to five lines of calculation.  It is the explanation I have made during the solution that has greatly increased the space used.

Solution 2: The Mathematical Approach

(a) First we write down the two vector equations of motion for the masses.

For mass X:    T – mxg = mxa   - (1)

For mass Y:    myg – T = mya   - (2)

Now, we solve these equations simultaneously.  So, adding equation 1 and 2 together we have:

a  =  (myg – mxg) / (mx  + my ) = (24 x 9.8 – 10 x 9.8) / (10 + 24)

= 4.035 ms-2

(b) Then from either equation 1 or 2, we can calculate the value of T.

From equation 1:        T = (10 x 9.8) + (10 x 4.035) = 138.35 N

This second solution is probably the more mathematically pleasing to the eye.  For my liking though, the previous solution is the more physically intuitive method.  Both solutions are equally acceptable and in the end it’s really only the setting out that differs.  Suit yourself as to which one you use.  You will find the more mathematical approach is safer as the problems become more complex.

EXAMPLE 2: Three masses of 2kg, 4kg and 6kg are connected by three light inextensible strings, X, Y and Z as shown below.  The masses are supported from the roof of a lift of mass 1000kg.  The lift is accelerating downwards with a net acceleration of 3 ms-2.

Determine:

a)       The tension in string X.

b)       The tension in string Y.

c)       The tension in string Z.

d)       The tension in the supporting cable.

SOLUTION:

Note that since the lift is accelerating downwards at 3 ms-2, we can write for any string supporting a mass inside the lift or indeed for the supporting cable itself that:

T = m (g – a), see Case 2 in the notes above.

(a) Tension in X = 12 x (9.8 – 3) = 81.6 N upwards

(b) Tension in Y = 10 x (9.8 – 3) = 68 N upwards

(c) Tension in Z = 6 x (9.8 – 3) = 40.8 N upwards

(d) Tension in the supporting cable = 1012 x (9.8 – 3) = 6881.6 N

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# Circular Motion

An object moving  in a circular path at constant speed is said to be executing uniform circular motion.  Obviously, although the speed is constant, the velocity is not, since the direction of the motion is always changing.  It can be shown that for an object executing uniform circular motion (UCM), the acceleration keeping the object in its circular path is given by:

ac = v2/r

where ac is called the centripetal ("centre-seeking") acceleration, v = speed of the object and r = radius of the circular path.  As the name implies, centripetal acceleration is directed towards the centre of the circle.

Using the fact that force can be written as F = ma, the centripetal force, Fc, acting on an object undergoing UCM is given by:

where m = mass of the object.  This force is also directed towards the centre of the circle.

Example: A car of mass 900 kg moves at a constant speed of 25 m/s around a circular curve of radius 50 m.  Calculate the centripetal force acting on the car.  (11250 N, towards the centre of the circle)

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## WORK

A force applied to an object is often capable of moving that object through a certain distance.  Whenever this happens we say that work has been done on that object.

Work is a scalar quantity defined mathematically as:

where W = work done on object, F = force acting on object along the line of motion and s = net displacement of object caused by the force along the line of motion.

The SI unit of work is the joule (J).  1J = 1 Nm

EXAMPLE: Calculate the work done when an object is moved through a displacement of 20m north by a force of 10N north.

ANSWER: W = F.s and so W = 10 x 20 = 200J (Note that there is no direction, since work is a scalar quantity.)

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# ENERGY

Energy and work are closely related quantities.  An object can do work only if it has energy.  Energy, then, is the property of a system that is a measure of its capacity for doing work.  The amount of energy an object has is equal to the amount of work it can do.  Like work, energy is a scalar quantity with an SI Unit of the joule (J).

Energy has several forms: electric energy, chemical energy, heat energy, nuclear energy, radiant energy (ie EM radiation such as light), mechanical energy (eg kinetic energy) and sound energy (ie the kinetic energy of the vibration of the air).  In a closed system (ie one in which no mass enters or leaves), energy can neither be created nor destroyed, although it may be transformed from one form into another.  This is called the Law of Conservation of Energy.

# KINETIC ENERGY

Kinetic energy is the name given to the energy associated with a moving body.  It can be shown that the amount of kinetic energy possessed by a body is given by:

where m = mass of the body and v = velocity the body.

Moving vehicles have kinetic energy.  Consider a small car of mass 920 kg moving at a constant speed of 60 km/h (16.67 m/s).  The kinetic energy of the car can be calculated as:

Ek = 0.5 x 920 x 16.672

Ek = 1.28 x 105 J

To change the velocity of a moving body, or to set a body at rest into motion, a net force must be applied to it and work must be done on it.  The work done on the body is equivalent to the change in kinetic energy of the body.

# POTENTIAL ENERGY

Stored energy is called potential energy, since it has the potential to do work for us.  Examples of potential energy include: the energy stored in a stretched (or compressed) spring; the chemical energy stored in a car battery; the energy stored in the water in a damn above a hydroelectric power station; and the energy stored in the chemical bonds holding compounds together.

# ENERGY TRANSFORMATIONS

Energy transformations (changes) are an important aspect in understanding motion.  In a car battery, chemical potential energy is transformed into electrical energy.  In an internal combustion engine (such as a car engine) the chemical potential energy stored in petrol is transformed among other things into mechanical energy in the form of kinetic energy of motion.

When cars collide, various energy transformations take place.  Some of the kinetic energy (KE) of the cars is transformed into sound – we hear the collision.  Some KE is changed into radiant energy (light energy) – we see sparks fly as the wreckage scrapes along the ground.  Some of the KE is transformed into heat – the friction produced by metal scraping on metal and tyres gripping the road under heavy braking produces heat.  Some KE is transformed into potential energy of deformation – the car bodies are compressed, compacted and twisted during the collision and some of the KE is stored in the deformed wreckage.  In a worst-case scenario, where an explosion takes place, chemical energy stored in the fuel is converted into kinetic energy (and sound, heat, light) as parts of the wreckage are flung far and wide.

# MOMENTUM:

Everyday experience tells us that both the mass and velocity of an object are important in determining things like (i) how hard it is to stop the object or (ii) the effect the object has in a collision with another object.  An 85 kg man running at 5 m/s is a lot harder to stop than a 15 kg six year old child running at the same speed.  A 50 gram bullet fired from a rifle with a muzzle velocity of 500 m/s will do a lot more damage than an identical bullet thrown at the target by hand.

Isaac Newton spoke of the “quantity of motion” of an object.  Today we define the momentum of an object to be the product of mass (m) and velocity (v).

Momentum is a vector quantity with SI Units of kgms-1 (or Ns, since 1N = 1kgms-2).

Newton’s 2nd Law can be re-written as:

where Dp = the change in momentum of the object and Dt = the time taken for the change in momentum to occur.

This quantity Dp (the change in momentum) is given the name impulse.  Clearly, from the above equation, impulse, I, is defined as the product of force and time and has SI Units of Ns.  Impulse is a vector quantity.

# CONSERVATION OF MOMENTUM

According to Newton’s 1st and 2nd Laws of motion, there is no change in momentum without the action of a net external force.  Thus, if no net external force acts on a particular system, the total momentum of the system must be constant.  This is known as the principle of the conservation of linear momentum.

A system on which the net external force is zero is given a special name.  Such a system is called an isolated system. So, another way to express the principle of the conservation of linear momentum is to say that within an isolated system the total momentum is a constant.  This principle is applicable to many important physical situations.

## CONSERVATION OF MOMENTUM DURING COLLISIONS

One important physical situation to which the principle of the conservation of linear momentum is applicable is the case of collisions between bodies.  In such cases, if we assume that no external net force acts during the collision, we can say that the total momentum of the system before collision equals the total momentum of the system after collision.  This proves to be an extremely useful starting point for analysing many collision situations.

To see that momentum is conserved during collisions we can use Newton’s 3rd Law.  Consider a collision between two particles, A and B, as shown below.

During the brief collision these particles exert large forces on one another.  At any instant FAB is the force exerted on A by B and FBA is the force exerted on B by A.  By Newton’s 3rd Law these forces at any instant are equal in magnitude but opposite in direction.

The change in momentum of A resulting from the collision is:

in which the bar above the FAB indicates that we are taking the average value of FAB during the time interval of the collision, Dt.

The change in momentum of B resulting from the collision is:

in which the bar above the FBA indicates that we are taking the average value of FBA during the time interval of the collision, Dt.

Note that it is necessary to take the average value of the collision forces since the magnitudes of both forces will vary over the duration of the collision.

If no other forces act on the particles, then DpA and DpB give the total change in momentum for each particle.  But we have seen that at each instant:

FAB = - FBA

So that:

And therefore that:     DpA = - DpB .

If we consider the two particles as an isolated system, the total momentum of the system is:

P = pA + pB

And the total change in momentum of the system as a result of the collision is zero, that is:

DP = DpA +  DpB = 0.

Thus, using Newton’s 3rd Law and our knowledge of impulse we have shown that if there are no external forces, the total momentum of the system is not changed by the collision.  Therefore, as we said before, if we assume that no external net force acts during the collision, we can say that the total momentum of the system before collision equals the total momentum of the system after collision.

How accurate is it though to assume that no external net force acts on a system during a collision?  When a golf club strikes a golf ball surely there are external forces that act on the system of club + ball?  Indeed there are: gravity and friction are two obvious forces that act on both club and ball.  So how can we simply ignore these forces?

The answer is that it is safe to neglect these external forces during the collision and to assume that momentum is conserved provided, as is almost always the case, that the external forces are negligible compared to the impulsive forces of collision.  If the external forces are negligible compared to the impulsive forces, then the change in momentum of a particle during a collision arising from an external force is negligible compared to the change in momentum of that particle arising from the impulsive force of collision.

In the case of the golf club striking the golf ball, the collision lasts only a tiny fraction of a second.  Since the observed change in momentum is large and the time of collision is small, it follows from the impulse equation:

Dp = F Dt

that the average impulsive force F is relatively large.  Compared to this force, the external forces of gravity and friction are negligible.  During the collision we can safely ignore these external forces in determining the change in motion of the ball; the shorter the collision time, the more accurate this assumption becomes.

In practice, we can apply the principle of momentum conservation during collisions if the time of collision is small enough.

*NOTE: This section of notes on Conservation of Momentum During Collisions has followed the treatment on pages 213-214 in “Physics Parts I & II Combined” by D Halliday & R Resnick (Wiley, 1966).

EXERCISE: Try out the "Hollywood Physics" exercise on the Brainteasers page.

FOR FUN - Check out this video on YouTube.

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# INERTIA

As we have seen, Newton’s 1st Law states that an object in uniform motion will remain in uniform motion and an object at rest will remain at rest, unless acted upon by an external, net force.  This ability of a body to resist changes in its state of motion is called the inertia of the body.  The inertia of a car, for instance, is its tendency to remain in uniform motion or remain at rest.  The fact that bodies possess inertia has important consequences when dealing with moving vehicles.

A moving vehicle is a complex body.  It consists of the vehicle body itself, the driver (and passengers) and other objects carried in the car.  If the driver or passengers or other objects are not restrained, they will continue to move at whatever speed the car is travelling, even after the application of the brakes.  Let us consider some questions:

u     What are some of the dangers presented by loose objects in vehicles?

u     Most people, when they see an unrestrained object fly off the car seat when they slam on the brakes, would say that it got pushed off the seat (ie some sort of force acted on the object to move it off the seat).  Is this an accurate account of the physics of the situation?  Explain in terms of Newton’s 1st Law.

u     Why do you think Newton’s 1st Law of Motion is not applied correctly in many real world situations (like the one mentioned above)?

u     What is the function of an inertia reel safety belt?  Where could we obtain information on how they work and their effectiveness?

u     Describe a possible experiment we could do to assess the relative effectiveness of lap, lap-sash and harness seatbelts in reducing the effects of inertia in a collision.

u     List the features of a modern car that are designed to reduce or avoid the effects of a collision.  Where would we obtain information to use in assessing the relative effectiveness of these features?

u     Assess the reasons for the introduction of low speed zones in built-up areas and the addition of air bags and crumple zones to vehicles with reference to the concepts of impulse and momentum.

JUST FOR FUN: Try this link for an interesting video showing a cinder block broken on top of a man lying on a bed of nails.

Or check out this one - .

See http://www.science.org.au/nova/058/058key.htm for a very interesting article on the physics of speeding cars based on work done by a statistician.  You may like to compare this article with this one at http://www.ibiblio.org/rdu/speedsci.html which was written by a theoretical physicist.  The second article uses more appropriate equations than the first and is probably more accurate in its estimates of risk.  Interesting, isn't it?  Please realize that this second article is not advocating speeding for the sake of speeding.  It is attempting to provide realistic estimates of the risks associated with speeding.  It presents one possible analysis.  There are others, as the article itself points out.

In my opinion, when on the roads, stick to the speed limits, obey all the road rules & concentrate on what you are doing.  This applies no matter how old you are or how experienced a driver you happen to be BUT especially if you are a young person, who has just started driving!!!

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