A force has components in the x and y directions respectively of 2 N andDN. Express this in unit vector form (i.e. ai + bj). The force acts on an object which moves from (1,5) to (6,E). Express the displacement in the formci + dj and hence determine

Task 1

You are testing an electronic circuit, and you discover that there are two voltage sources in separate parts of the circuit. The voltages can be modelled by the following equations:

V₁= A sin ( 100 + /9 )

V₂= B sin ( 100t²/5 )

i) To investigate these voltages, you are required to determine the following parameters: amplitude, phase angle and frequency, and the periodic time of each of these waves.

ii) The power sources are switched on simultaneously. How many seconds does it take for each respective voltage to reach its maximum?

iii) You believe that there is a link that is combining the voltages V₁and V₂. As a result, you would like to determine the nature of this combined waveform. Use appropriate compound angle formulae to expand the expressions for both V₁ and V₂into the form:

V_a = M sin⁡〖100t & N cos⁡100t 〗

where M and N are real numbers and V_a= V₁or V₂. You should determine the values of M and N for both expressions.

iv) Using your solution from (iii), express the resultant of V₁+ V₂in the form:

V_res = R sin⁡(100t)

where R = amplitude of the resultant and = phase angle

v) Your manager would like a graphical illustration of the addition of the voltages and has asked you to use appropriate software to plot V₁and V₂. With the aid of the software, plot the resultant of V₁+ V₂over two cycles to confirm the analytical solution by finding the amplitude and phase angle.

Task 2

The top of a support post in the workshop is acted on by three forces: F1, F2 and F3

F1 has a magnitudeCand acts in the direction (1, 2, 2)

F2 has a magnitude 3 and acts in the direction (1, 2, -2)

F3 has a magnitude 4 and acts in the direction (3, -2, 3)

Your manager is concerned about the stability of the support and has asked you to investigate. Determine the resultant force acting on the top of the support. Confirm your analytical result using suitable existing formulae in mathematical software such as Excel.

Task 3

A force has components in the x and y directions respectively of 2 N andDN. Express this in unit vector form (i.e. ai + bj). The force acts on an object which moves from (1,5) to (6,E). Express the displacement in the formci + dj and hence determine the work done using the dot product (Note: Work done = Force*Displacement). The unit of displacement is 1 metre. Confirm your analytical result by using a suitable existing formula in mathematical software such as Excel.

Recommended Resources

Please note that the resources listed are examples for you to use as a starting point in your research the list is not definitive.

Course notes for Engineering Mathematics LO3 on UCN VLE

Bird J. (2021)Higher Engineering Mathematics. 9^thEd. Routledge

Stroud, K.A. and Booth, D.J. (2020)Engineering Mathematics8^thEd. Bloomsbury Publishing

Learning Outcomes and Assessment Criteria

Pass	Merit	Distinction
LO3Use analytical and computational methods for solving problems by relating sinusoidal wave and vector functions to their respective engineering application.		D2 Apply engineering mathematical software to confirm the analytical solutions for at least three engineering/manufacturing problems involving sinusoidal and vector functions.
P6 Solve engineering/manufacturing problems relating to sinusoidal functions. P7Use appropriate methodology to determine engineering parameters of data represented in vector form.