ENG1014: ENGINEERING NUMERICAL ANALYSIS
LAB 6 – WEEK 6
2023 S1
Welcome to lab 6. Remember that laboratories continuously build on previously learned concepts and lab tasks.
Therefore, it is crucial that you complete all previous labs before attempting the current one.
Self-study:
Students are expected to attempt these questions during their own self-study time, prior to this lab session. There may
be questions that require functions not specifically covered in the workshops. Remember to use MATLAB’s built-in help
for documentation and examples.
Learning outcomes:
1. To identify kinetic, gravitational potential and non-conservative energy sources.
2. To utilize the conservation of energy equation to solve dynamical problems
3. To apply Matlab skills to solve conservation of energy problems.
4. To apply ‘for’ and ‘while’ loop skills to a Physics and Energy context
Submission requirements:
The lab submission links can be found on Moodle under the weekly sections. The submission box will only accept one
.zip file. Zipping general instructions are described here.
Deadline:
Officially, the lab is due on Thursday 6th April at 11:55pm AEST. However, the late penalty will only apply after Friday 7th
April at 11:55pm AEST.
Lab 6 – Assessed questions
Remember good programming practices for all tasks even if not specifically stated. This includes, but is not limited to:
● using clc, close all, and clear all, where appropriate
● suppressing outputs where appropriate
● labelling all plots, and providing a legend where appropriate
● fprintf statements containing relevant answers
TASK 1
[2 MARKS ]
Note: Team tasks are designed for students to recall material that they should be familiar with through the workshops
and to recall knowledge gained from individual exercises prior to this lab session.
Conservation of energy is an important concept throughout all streams of engineering. Understanding the transfer of
energy from one mode to another helps provide insight into systems and how they function.
A mass of 5 ?? is connected to a wall via a spring with a spring constant of ? = 50 ?/?. Assume no external forces
such as friction or drag act on the mass, and assume that the mass is released from rest at ? = 0? from ?
0 = 0. 2 ?
(0. 2 ? to the right of the equilibrium position of ? = 0 ?).
1. Draw a free body diagram of the system initially when . ?
0 = 0. 2 ?
2. Write the second order differential equation for the system in terms of ? metres at time ? ???????.
3. Sketch a plot indicating approximately how ? changes with time.
2 Important: If you are struggling with a task, ensure that you have performed hand-written work (e.g. hand
calculations, pseudocode, flow charts) to better understand the processes involved. Do this before asking
demonstrators for help and use it to assist with your illustration of the problem.
The same system is now rotated to have the 5 kg mass hanging vertically from a ceiling via the spring with spring
constant of . A damper is also added to the system with . Assume that , ? = 50 ?/? ?
? = 10 ??/? ? = 9. 81 ?/?
2
? below the equilibrium position and the object is released from rest. It is known that the damper creates an
0 = 0. 2 ?
underdamped system.
4. Draw a free body diagram of the system initially when below equilibrium position. ?
0 = 0. 2 ?
5. Determine (the change in equilibrium position relative to the anchor point of the spring, from ∆(? = 0)
situation 1 to situation 2).
6. Write a second order differential equation for the system in terms of ? metres at time ? ???????.
7. On the same axis as the horizontal system, draw a plot showing an estimation of how ? changes with time.
8. Comment on similarities and differences between the system when it is horizontal versus vertical.
3 Important: If you are struggling with a task, ensure that you have performed hand-written work (e.g. hand
calculations, pseudocode, flow charts) to better understand the processes involved. Do this before asking
demonstrators for help and use it to assist with your illustration of the problem.
TASK 2
[2 MARKS]
R2-D2 (R2) is resting against a spring under compression with a spring constant of 85. 0 ?/?. R2 and the spring is on a
table above the ground. R2 has a mass of . Assume no friction or drag, and . fprintf 0. 750 ? 2. 50 ?? ? = 9. 81 ?/?
2
your results to 3 decimal places for each part and briefly comment on your working.
a) R2 is released from the compressed position (point A) and moves along the table, reaching 6 ?/? at equilibrium
position of the spring (point B). What distance does R2 travel before the spring reaches its equilibrium length?
b) After point B, R2 continues to roll and fall off the table. At what vertical speed does R2 hit the floor?
c) Hence, what is the total kinetic energy of R2 when he hits the ground?
d) If R2 hits the floor with a vertical velocity of more than 5 m/s, he breaks into pieces. The table is a standing desk,
and can increase its height by 1cm at a time. How high can the table be before R2 breaks?
4 Important: If you are struggling with a task, ensure that you have performed hand-written work (e.g. hand
calculations, pseudocode, flow charts) to better understand the processes involved. Do this before asking
demonstrators for help and use it to assist with your illustration of the problem.
TASK 3
[2 MARKS]
R2-D2, always being the prankster, pushes BB-8 down the 1.50m long Millennium Falcon ramp. When he reaches the top
of the ramp, BB-8 has a velocity of . BB-8 is , and the ramp is at a angle to the ground. Assume 4. 80 ?/? 1. 30 ?? 27. 0
?
that ? = 9. 81 ?/?
2
.
a) Calculate the height and gravitational potential energy of BB-8 when he is on the Falcon, and use a fprintf()
statement to print the result to 3 decimal places.
b) Calculate the total velocity and kinetic energy of BB-8 when he reaches the ground, and use a fprintf() statement
to print the result to 3 decimal places. Assume no friction.
c) BB-8 actually enjoys rolling down the ramp and wants R2 to change the angle to the ground to a range of
different angles to experience different energy transfers. For angles between 15 ˚ to 45 ˚, in increments of 5˚,
produce the following on one figure:
o Plot height (m) against angle (˚ ) on a conventional left axis.
o Plot velocity (m/s) at the bottom of the ramp against angle ( ˚ ) on the right axis.
5 Important: If you are struggling with a task, ensure that you have performed hand-written work (e.g. hand
calculations, pseudocode, flow charts) to better understand the processes involved. Do this before asking
demonstrators for help and use it to assist with your illustration of the problem.
TASK 4
[2 MARKS]
Andrew drops a ball from the top of the Woodside Building from a height of 40m above ground level. Assume that g =
9.81 m/s^2 and the ball is 1kg.
For now, assume that the collision is elastic (i.e., e = 1).
a) Determine the kinetic energy from h = 40m to h = 0m (from the top to ground level) Repeat the same for
potential energy. Use either anonymous functions or a vector with resolution of t = 0.01 seconds.
b) Determine the kinetic energy from h = 0 m to the peak height after bouncing. Repeat the same for potential
energy. Use either anonymous functions or a vector with resolution of t = 0.01 seconds.
c) Use your results from above to create a vector of kinetic energy values for 5 bounces (finishing at the peak
height after the 5th bounce) with a resolution of t = 0.01 seconds. Repeat this process for potential energy and
plot both energy vectors against time on the same graph.
d) Comment with fprintf on how the graph would look different if the collision was inelastic (i.e., e < 1).
6 Important: If you are struggling with a task, ensure that you have performed hand-written work (e.g. hand
calculations, pseudocode, flow charts) to better understand the processes involved. Do this before asking
demonstrators for help and use it to assist with your illustration of the problem.
TASK 5
[2 MARKS]
A system is shown below where a hanging mass is attached to a spring, and is allowed to freely ‘bob’ up and down
(oscillate).
a) The mass of the object is 5 kg, the spring constant is and . Assume that the mass is 50 ?/? ? = 9. 81 ?/?
2
released from rest at at from the equilibrium position and assume there is no damping. ? = 0 ? ?
0 = 1. 2 ?
(i) Define the equation for ?(?), the position of the mass relative to the equilibrium position as an anonymous
function.
(ii) Plot the graph of ?(?) against t from 0 to 5 seconds with a resolution of 0.01.
b) The same situation is created, except now we add a damper to the system, where is the damping ?
? = 10 ??/?
constant. Still assume that below equilibrium and . ? = 5 ??, ?
0 = 1. 2 ? ? = 50 ?/?
(i) Determine and use fprintf to specify the classification of the system (undamped, underdamped, critically
damped or overdamped).
(ii) Hence, define the new equation for using below equilibrium and . ?(?) ? = 5 ??, ?
0 = 1. 2 ?
? = 10 ??/?
(iii) Plot ?(?) against ? for 0 to 5 seconds with a resolution of 0.01 seconds.
7 Important: If you are struggling with a task, ensure that you have performed hand-written work (e.g. hand
calculations, pseudocode, flow charts) to better understand the processes involved. Do this before asking
demonstrators for help and use it to assist with your illustration of the problem.
c) How long does it take for the mass to travel a total distance of 2 metres with the new system? Use a step of 0.01
seconds when looping through time. fprintf this answer to 2 decimal places.
2 marks deducted for poor programming practices (missing comments, unnecessary outputs,
no axis labels, inefficient coding, etc.)
END OF ASSESSED QUESTIONS
The remainder of this document contains supplementary and exam-type questions for extended learning. Use your
allocated lab time wisely!
8 Important: If you are struggling with a task, ensure that you have performed hand-written work (e.g. hand
calculations, pseudocode, flow charts) to better understand the processes involved. Do this before asking
demonstrators for help and use it to assist with your illustration of the problem.
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