International Study Centre
Applied Maths 1 – July 2020
Final Coursework Assessment Resit – 4 hours
This assessment is 100% of the module total
INSTRUCTIONS
There are 10 questions in this assessment. Answer all questions. You must show all working
to get full marks. Please write in blue or black ink. Candidates are reminded of the need for
good English and orderly presentation of their answers. Remember to include units when
necessary.
ONLY APPROVED CALCULATORS ALLOWED
For markers’ use only:
1st marking
2nd marking
Question 1
/7
/7
Question 2
/7
/7
Question 3
/8
/8
Question 4
/8
/8
Question 5
/ 13
/ 13
Question 6
/ 13
/ 13
Question 7
/ 11
/ 11
Question 8
/ 11
/ 11
Question 9
/ 11
/ 11
Question 10
/ 11
/ 11
Total
/ 100
/ 100
Authored by: Tom Bossingham
Internally moderated by: Dr Roberto Sisca
Externally moderated by: Prof. Tom Crick
Page 1 of 11
Date:
Date:
Date:
July 2020
July 2020
July 2020
Question 1:
Consider the vector r¯ stated below,
r¯ = −2ˆi + 6ˆj.
a) Find the unit vector rˆ of the vector stated above.
[4 marks]
b) Explain, using words, how you would find an expression for the
vector R¯ that is parallel to the vector r¯ but has a length |R¯|.
[The answer to this part of the question must be typed in own words.]
[3 marks]
Page 2 of 11
Question 2:
Two vectors a¯ and ¯b are expressed below in the unit vector notation,
a¯ = 7ˆi − 3ˆj + 7kˆ,
¯b = 3ˆi + 4kˆ.
a) Calculate the angle θ between the two vectors a¯ and ¯b.
[4 marks]
b) What can you infer about the scalar product ˆi · ˆj. Make sure you justify your
answer using words.
[The answer to this part of the question must be typed in own words.]
[3 marks]
Page 3 of 11
Question 3:
A ball falls off a ledge and moves vertically downwards under the
influence of gravity towards the ground. The ledge is positioned at a height
of h = 9.70 m above the ground.
a) Calculate the time t required for the ball to reach the ground.
[4 marks]
b) If the initial height of the ledge was doubled (h → 2h), does the amount of time
required for the ball to fall to the ground also double? Explain your answer using
words. Comment on the factor increase in the time taken to fall a height of 2h
when compared to h.
[The answer to this part of the question must be typed in own words.]
[4 marks]
Page 4 of 11
Question 4:
An object starts at an initial position and is moving horizontally on a straight line
with an initial velocity of u = 19ˆi ms−1. The object experiences an acceleration
in the opposite direction to its initial motion, the magnitude of this acceleration
is |a¯| = 4.90 ms−2.
a) Express the acceleration as a vector in terms of the unit vector ˆi.
Explain, using words, the sign convention and what it represents.
[The answer to this part of the question must be typed in own words.]
[4 marks]
b) Calculate the distance between the initial position of the object and the
position at which the object comes to rest.
[4 marks]
Page 5 of 11
Question 5:
A mass of m = 2 Kg is positioned on a smooth incline at an angle of elevation of
θ = 32o above the horizontal. The mass is at a distance d from the bottom of the
incline. see diagram below.
𝑚
d
𝜗
a) Calculate the acceleration of the mass down the incline.
[4 marks]
b) Calculate the reaction force that the surface exerts on the mass.
[3 marks]
Now consider the surface of the incline to be rough such that the
coefficient of friction is non-zero µ = 0.
c) Explain, using words, how a rough surface will affect the acceleration
experienced by the mass.
[The answer to this part of the question must be typed in own words.]
[3 marks]
d) Calculate the value of the coefficient of friction µ needed for the mass to be
in limiting equilibrium.
[3 marks]
Page 6 of 11
Question 6:
A box of mass m = 5 Kg is on a smooth horizontal surface with a force F¯
acting on it at an angle θ = 40o above the horizontal. See diagram.
𝐹
𝜗
𝑚
a) If the horizontal acceleration of the box is ax = 7 ms−2, find the
magnitude of the pulling force F¯.
[4 marks]
b) Calculate the corresponding reaction force R that the horizontal
surface exerts on the box.
[3 marks]
c) Explain, using words, how your consideration of Newton’s third law helped
you to calculate the reaction force that the surface exerts on the mass in
part b).
[The answer to this part of the question must be typed in own words.]
[3 marks]
Consider the surface to now be rough with a coefficient of friction given by
µ = 0.30.
d) Calculate the new horizontal acceleration of the box.
[3 marks]
Page 7 of 11
Question 7:
A mass m1 = 14 Kg is initially at rest on a smooth horizontal surface. The
mass m1 is attached to a light in-extensible string which runs over a smooth pulley
and is attached to another mass m2 = 4 Kg that hangs ver- tically downwards.
See diagram.
𝑚1
𝑚2
a) Use Newton’s second law to find an equation of motion for each of the
masses.
[4 marks]
b) Solve the two equations of motion simultaneously to find the the value of the
tension T in the string and the acceleration a of the masses.
[4 marks]
c) Calculate the time required for the mass m2 to fall a vertical distance of
1.80 m given that the mass was initially at rest.
[3 marks]
Page 8 of 11
Question 8:
A mass m2 = 5 Kg is attached to a light in-extensible string that passes
over a smooth fixed pulley and under another smooth pulley of mass
m1 = 2 Kg. Both masses are free to move vertically and initially the system is at
rest. When the masses are in motion the mass m1 accelerates vertically
upwards whilst the mass m2 accelerates vertically downwards. See diagram.
𝑚1
𝑚2
a) Use Newton’s second law of motion to find an equation of motion for
each of the masses.
[4 marks]
b) Solve the two equations of motion simultaneously to calculate the
acceleration of the mass m2.
[4 marks]
c) Given that the masses were initially at rest, calculate the velocity of the
mass m2 after it has been in motion for t = 3.10 s.
[3 marks]
Page 9 of 11
Question 9:
A mass of m = 9 Kg is sliding in a straight line across a rough horizontal surface
with a coefficient of friction µ = 0.50. The initial velocity of the mass is v =
9 ms−1 and it eventually comes to rest at a distance d from its initial position.
a) By considering work-energy equivalence, calculate the work done by
friction in bringing the mass to rest.
[3 marks]
The work done by a constant force is given by,
W = F d,
where W is the work done, F is the magnitude of the constant force and d
is the distance moved.
b) Explain, using words, why the constant force due to friction in this case is equal
to Fmax, the maximum frictional force that can be provided by the surface.
[The answer to this part of the question must be typed in own words.]
[4 marks]
c) Use energy conservation to calculate the distance d.
[4 marks]
Page 10 of 11
Question 10:
A mass of m = 8 Kg is sliding across a smooth horizontal surface with a velocity v
= 12 ms−1. The mass collides directly with a horizontal spring and causes
the spring to compress until the mass comes to rest. The value of the spring
constant is k = 217 Nm−1.
a) Explain, using words, how energy conservation can be used to
calculate the compression of the spring.
[The answer to this part of the question must be typed in own words.]
[4 marks]
b) Calculate the initial kinetic energy of the mass.
[2 marks]
The elastic energy stored in a spring is given by,
1
Eelastic = kx 2,
2
where Eelastic is the elastic energy, k is the spring constant and x is the
compression/extension of the spring.
c) Calculate the compression of the spring needed to bring the mass to
rest.
[5 marks]
END OF
EXAM.
Page 11 of 11
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