I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.
Week 12
We finished the section on integrals related to the inverse trigonometric functions. We revisited work and then spoke about centroids of laminas and centres of gravity of volumes of revolution.
Week 13
I will be available to any and all students at the following times and venues:
- Review Lecture Tuesday 15:00 B260
- Review Lecture Wednesday 11:00 B214
- Review Lecture Thursday 11:00 B260
- Review Tutorial Friday 11:00 B260
- Review Tutorial Friday 13:00 B245
The Review Lectures will be conducted as follows
- Students can ask any question and I will answer it on the whiteboard. If we run out of questions
- I will start going through the Winter 2013 paper (which was given out in class). If we finish this paper
- I will help ye one to one.
The Review Tutorials will be conducted as follows
- Students can ask any question and I will answer it on the whiteboard. If we run out of questions
- I will help ye one to one. You can work on whatever you want but I will be recommending a centroids and centres of gravity.
For your study, I would strongly recommend that you attempt this paper before the review lectures. This is your best chance of seeing how you stand.
Exam Format
Four questions, do all as in the Winter 2013 paper which can be found by searching through Mathematics.
Academic Learning Centre
Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.
Math.Stack Exchange
If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.
Additional Notes: E-Books
If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.
J.P. McCarthy: Math Page 
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August 16, 2014 at 3:31 pm
Student
Hi J.P.,
I am stuck on these vector questions that came up in the summer exam!
Question 2 (a) (iii), (b) and (c)
Thanks a million for your help.
August 16, 2014 at 3:58 pm
J.P. McCarthy
In the below vectors are represented in BOLD. Obviously you can’t do this with pen and paper so your vectors should be underlined; e.g. instead of
not 
but 
.
For (a) (iii) we need to know the theorem
This reads, “vectors
and 
are PERPENDICULAR if and only if the their dot product is zero.”
If and only means that the two statements are equivalent and the theorem actually says two things:
“IF the vectors are perpendicular, THEN their dot product is zero”, and
“IF the dot product of two vectors is zero, THEN they are perpendicular”.
To say the two statements — ‘the vectors are perpendicular’ and ‘the vectors have zero dot product’ — are equivalent means that whenever you hear one you can think of the other and vice versa.
So in our question here, if we are to suppose that ‘
is perpendicular to 
‘ then by the above discussion this just means that
So to ask for the value of
that makes the vectors perpendicular is the same as asking for the value of 
that makes (*) hold.
So we solve for
:
Now calculating dot products is one of the easier things to do in this module:
I will let you finish this off.
Part (b) concerns WORK. We have that the work,
, done by a force, 
, in moving on object through a displacement 
is given by
.
Here we are told that the displacement is
and the force is 
.
Can you finish this off if I told you to watch the units?
Part (c) concerns the moment or TORQUE of a force. A force can have a turning effect and this turning effect is the moment or torque,
. When the force, 
, acts at a displacement from the moment axis, 
, then the torque is given by
where ‘
‘ is the VECTOR or CROSS PRODUCT.
At this point I can’t explain much without a picture and I would encourage you to look your notes.
Briefly the displacement is the vector that goes from the turning point
to the point where the force acts 
and
The calculation of the cross product is also covered in your notes.
Regards,
J.P.