1st Year Mechanics
Modelling Workbook
Modelling is defined in this lecture course as being a series of representations used to
reach the solution of a given problem.
Representations can be diagrammatical,
graphical, mathematical and/or verbal. The aim of this workbook is to provide practice
at the various representations. This should be useful not only in mechanics but in other
areas of physics too. Solutions will be provided as you work through this booklet.
However, in order to make the most of this resource, you are encouraged to complete
the exercises in your own time before Dr Sands works through the examples in the
lectures. Not everyone will find these examples easy, if you get stuck...ASK!
Diagrams are used to represent the agents or objects in a given problem and the
relationships between them. In mechanics, diagrams are most often used to make sense
of the forces acting on objects. Some of the following examples require you to construct
a simple diagram from a written description. These problems may seem simple, but
drawing diagrammatical representations is not trivial when the problems become more
complicated. Practice is the key. Not all the examples are mechanics based, but
hopefully this will help your confidence in using diagrams in other areas of physics, too.
It is important that when you drew diagrams, you include as much of the information
given in the description as possible. You might have to add information not included in
the description as well, using your knowledge of general physics (e.g. air resistance etc).
Graphical representations are used to analyse situations in more detail from
information given in a problem, for example to find convergence points, intercept
positions, equilibrium points.
Mathematical representation is the one which most students find the easiest ,as
numbers are usually plugged into equations and the required answer comes out at the
end. In this case, however, the idea is to be able to manipulate applicable equations to
represent a problem, which is just as important as reaching a final numerical solution.
Mathematical representations can be used to great effect in conjunction with
diagrammatical and graphical, as a lot of information can be included in a model using
these three ways of representing what you know.
Verbal representations are there to demonstrate what you know about a problem and
the solution. You should be able to describe the model you have constructed and
representations you have made. In fact, verbal representations are happening from the
minute you start analysing a problem as you have to make sense of what is written in
order to represent the information in the different ways mentioned above. You can use
the verbal description to explain the assumptions you have made, conclusions you have
drawn and results you have found from your other representations.
Question 1
A boy throws a bouncy ball straight up in the air. Draw a simplified diagram of the path
of the ball from the point it leaves the boys hand to the point it eventually comes to rest
on the floor. On this diagram, label the forces acting at each point.
Question 2
A cart rolls down a frictionless ramp. Draw a simple diagram to represent this scenario.
Take care to show all the forces acting on the cart.
Question 3
Draw a simple bar magnet showing the field lines. Now draw two bar magnets end to
end with the S poles a small distance apart. What would the field lines look like now?
What about if the N and S poles were end to end?
Question 4
A man is swimming across a fast flowing river.
a) Draw a diagram of this problem and the forces acting on the man as he swims
across.
b) There is only one point on the other side where he can safely climb out which is
directly opposite the point at which he dives in. Draw a diagram showing the
path he must take in order to be able to climb out assuming he can maintain a
constant strong swimming speed.
Question 5
Consider a horse-shoe shaped magnet. A loop of wire carrying a current is positioned
such that it is in the magnetic field. Draw a diagram representing the system and the
forces acting on the wire (Hint: ‘Flemming’s right hand rule’.) What would the result be
on the motion of the wire?
Question 6
A woman and her infant son are sitting on a train travelling at a speed of 160km/hr. She
throws her baby in the air and then (luckily) catches him. Draw a representation of the
path of the baby from the point of view of a person standing on the platform as the train
goes through the station.
Question 7
Consider a pendulum consisting of a small ball on the end of a string. At the equilibrium
position of the system and positioned halfway up the string is a razor blade placed to
cut the string. The pendulum is set to swing from an arbitrary angle, θ.
a) Draw a simple diagram of the system just before the pendulum is set in motion.
b) Draw a diagram of the path of the ball after the string has been cut and label the
forces on the ball.
c) Verbally describe the path of the ball from the point just after the string is cut.
(Hint: describe the forces acting on the ball to explain the path you have chosen.)
Question 8
A ball rolls down a slope and hits a wall at the bottom with an imperfect elastic collision.
Draw a diagram of this scenario. Draw a graphical representation of the speed of a ball
from the start of the roll to the point at which it comes to rest.
Question 9
A cyclist crosses a line on the road travelling at 5km/hr and travels at this speed for 5
minutes. He then increases his speed by 1 km/hr and does so every 5 minutes after
crossing the line. Use a graphical representation, determine at what speed will he cross
a second line on the road 10km away from the first. Assume that he can increase his
speed instantaneously without any ramp up time.
Question 10
A teabag on a string is dipped into a cup of boiling water. Does the cup get heavier if the
string is still taut?
Use a diagram and mathematical representation to solve this
problem.
Question 11
A negatively charged rod is placed near a thin stream of water.
Use a verbal
representation to describe the result. What would happen if the rod were positively
charged instead and why?
Question 12
A U shaped tube is suspended such that the two openings are at the top. A ball is
dropped into each end, one is twice as heavy as the other. Verbally describe the result
of dropping the balls at the same time.
Question 13
An airplane accelerates down a runway with an acceleration of 3.2m/s2 for 32.8s until it
lifts off. Use a mathematical model to determine the distance travelled before takeoff.
Question 14
You are driving and see a bridge in the distance. Your guidebook says that the bridge is
192m high and you estimate the angle from where you stand to the top of the arch to be
approximately 2˚. Use diagrammatical and mathematical representations to determine
your approximate distance to the base of the bridge.
Question 15
A ball is thrown straight up in the air and is caught in the same position 2s later. Use
mathematical and diagrammatical representations to determine the initial speed of the
ball and the maximum height the ball reaches.
Question 16
A block of mass 0.5kg slides on a flat smooth surface with a speed of 2.80m/s. It then
slides over a rough surface with frictional coefficient μk and slows to a halt. While the
block is slowing, (a) what is the frictional force on the block? (b) What is the magnitude
of the block’s deceleration? (c) How far does the block slide on the rough part before it
comes to a stop? (d) Does the deceleration depend on mass? Use mathematical
representations to answer this question.
Question 17
A 3kg mass slides on a smooth horizontal surface; it is joined by a string to a hanging
2kg mass. The string passes over an ideal pulley. When the 2.00 kg mass is released,
what is the acceleration of the two masses? What is the tension in the string? Use
Diagrammatical and mathematical representations should be used for this question.
The following questions aim to bring together the representations you have been
practicing in order to build complete models to solve problems.
Question 18
Batman jumps from a multi-storey car-park onto a van driven by a criminal. The
distance between the ledge he is standing on and the roof of the van is 10m. Batman
weighs 100kg and his armour weighs 20kg. He forgets to put his wings out when he
jumps because he hasn’t been sleeping too well and Alfred gave him decaffeinated
coffee by accident.
Use mathematical, graphical AND diagrammatical methods to
represent the problem and determine the velocity at which Batman lands on the van.
Use a verbal representation to describe the solution and make comments as to whether
this problem is realistic.
Question 19
We always see the same side of the moon. Using at least two forms of representation (I
recommend verbal and diagrammatical) explain why this is the case.