Higher National Assessment

Higher National Assessment Designed in accordance with HEA guidelines

Learner Name: Year of Course 1

Programme Title Pearson BTEC Level 4/5 Higher National Certificate/Diploma in Engineering (RQF)

Programme No BBYH8 – BBYH9 – BBYJ1 DBMT1 – DBMT4

Unit Title 2. Engineering Maths (L4) Unit Code M/615/1476

Assignment Title Analytical & Computational Methods Assignment No 2 of 3

Author Jamie Caulfield Assessor Engineering Team Internal Verifier Michael Lopez Verification Date 2nd April 2021

Week of Issue Due Week Date Submitted Agreed Resubmission Date Date Resubmitted 21 24

Assessor feedback to Learner: – Formative/Summative Grade

I certify that the evidence submitted for this assignment is my own. I have clearly referenced any sources used in the work. I understand that false declaration is a form of malpractice. Learner Email Address: Date:

I certify that the evidence submitted for this assignment is the learner’s own. The learner has clearly referenced any sources used in the work. I understand that false declaration is a form of malpractice. Assessor Signature: Date:

Note: All the above information should feature on the cover page.

Submission Format You should submit: – A series of hand-written and/or word-processed responses for all the given tasks.

Relevant Learning Outcomes and Assessment Criteria Pass Merit Distinction

LO3: Use analytical and computational methods for solving problems by relating sinusoidal wave and vector functions to their respective engineering application

D2 Model the combination of sine waves graphically and analyse the variation in results between graphical and analytical methods.

P6 Solve engineering problems relating to sinusoidal functions.

M3 Use compound angle identities to separate waves into distinct component waves.

P7 Represent engineering quantities in vector form, and use appropriate methodology to determine engineering parameters.

Unit Learning Outcomes LO3: Use analytical and computational methods for solving problems by relating sinusoidal wave and vector functions to their respective engineering application.

Assignment Brief and Guidance Scenario: You work as a technical analyst for a large publisher of engineering text books. One of their well- known authors has set a number of further problems in her latest draft engineering text, aimed at the level four engineering market. She wishes to set these further problems throughout the text and intends to provide the worked solutions on a companion website for the new book. The author has asked you to provide these worked solutions. The further problems are given in the following tasks: – Task 1: Note: You need to have your calculator in radians (RAD) mode for this task (since the angles are given in radians – i.e. π is featured).

a) A current waveform may be described by…

is = 13 cos �2πft − π 4 � [A]

where frequency, f = 1Hz and t represents time.

Make time (t) the subject of this formula and hence determine a point in time when the current waveform has a magnitude of +10A.

b) The instantaneous value of a power signal may be described by;

12∠� 3π 8 � [W]

Find the magnitude of the vertical and horizontal components of this signal. Note that the symbol ∠ indicates ‘angle’. Task 2:

a) A resistor, R, is connected in series with an inductor, L. An a.c. current, 𝑖𝑖, flows through this RL combination, causing a voltage (𝑉𝑉𝑅𝑅) of 30V to be developed across the resistor, and a voltage (𝑉𝑉𝐿𝐿) of 40V to be developed across the inductor. Assuming that 𝑉𝑉𝑅𝑅 is in-phase with 𝑖𝑖, and 𝑉𝑉𝐿𝐿 leads 𝑉𝑉𝑅𝑅 by 90𝑜𝑜, draw a vector diagram for this arrangement and then calculate the magnitude of the resultant voltage across the whole RL combination.

b) A current-carrying filament is subjected to a strong magnetic field within an experimental chamber. It is required to find the force on the filament. The current may be modelled in three-dimensional space as:

𝐼𝐼 = 2𝑖𝑖 + 3𝑗𝑗 − 4𝑘𝑘

and the magnetic field as:

𝐵𝐵 = 3𝑖𝑖 − 2𝑗𝑗 + 6𝑘𝑘

Find the Cross Product of these two vectors to ascertain the characteristics of the force on the filament (i.e. find 𝐼𝐼 × 𝐵𝐵). Sketch this Cross Product (or use software to do so).

Task 3: PART 1 The two signals below are sensed by a signal processor;

𝑣𝑣1 = 40 sin(4𝑡𝑡)

𝑣𝑣2 = 𝐴𝐴 cos(4𝑡𝑡) The signal processor adds the signals to form a third signal, which must be described as a distinct signal in the following form;

𝑣𝑣𝑜𝑜 = 50 sin(4𝑡𝑡 + 𝛼𝛼) Use a compound angle identity to determine the value of A (the amplitude of 𝑣𝑣2). Ensure that you have your calculator in Radians (RAD) mode when determining your answer. Use graphical software to plot/model the inputs and output of the signal processor. How do you think graphical methods of sine wave combination compare with analytical methods? PART 2 The third harmonic of a sound wave is given by;

4 cos(3𝜃𝜃) − 6 sin(3𝜃𝜃) Express this sound wave in the form;

𝑅𝑅 sin(3𝜃𝜃 + 𝛽𝛽)