Term 2 Flashcards
(109 cards)
1
Q
What are the 9 Key Properties of Dot Product
A
2
Q
What are the 12 key properties of Cross Product
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3
Q
Find a vector orthogonal to
p = 3i -4j +2k
q= 2i+5j-k
A
4
Q
Find the angle between the vectors
p = i +2j +3k
q = 3i + 2j + k
A
5
Q
Find the work done by a force F = 5i -j + 2k , which acts on a body that undergoes a displacement i +2j + 2k m
A
6
Q
given
p = i +2j
b = j + 3k
c = 2i -k
show that a dot (b X c) and (a X b) dot c are equal
A
7
Q
How do you calculate tangential velocity of a rotating object
How do you calculate angular momentum of a rotating object
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8
Q
A
9
Q
Take integral over x variable
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10
Q
Take integral over y variable
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11
Q
How do you convert Cartesian to polar coordinates
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12
Q
How do you convert Cartesian to cylindrical coordinates
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13
Q
How do you convert Cartesian to spherical coordinates
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14
Q
A
15
Q
How do you convert spherical coordinates to cartesian
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16
Q
How do you convert cylindrical coordinates to cartesian
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17
Q
What is the formula to calculate area and volume in cylindrical coordinates
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18
Q
What is the formula to calculate area and volume in Spherical coordinates
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19
Q
A
20
Q
how is the gradient function expressed
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21
Q
Find the grad function of the following function
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22
Q
express the grad function in terms of unit vectors
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23
Q
Define the derivative of the velocity vector (first Principles)
Find the first and second derivatives to it
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24
Q
find the parametrised form of a circle of radius 4 and centre 2,3
A
25
derive the centripetal acceleration formula
26
find the potential difference between two points given by E = (2,-2,3) NC^-1 if the displacement from point A to B is given by s = (1,4,-2) m
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Find all unit vectors a, b, and c which fit in a plane such that
π+π=π
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Describe the transformations
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symbols for polar and azimuth have switched round
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Derive the Continuity Equation for a Compressible Fluid
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Using Matricies
40
Derive the Vector Equation for Gravitation
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Show how
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Calculate the Laplacian (Second derivative) of the gravitation scalar function
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What are the characteristics of conservative fields
44
Define the Curl of a Vector Field
In a conservative field, the vector field can be described as the gradient of a scalar potential function. Vector field at every point can be described using 1 vector pointing in the direction of the greatest rate of increase. Curl =0
In a non-conservative field, the vector field cannot be described using only a single vector. Hence rotation is observed
Curl is a vector of the greatest rotational influence
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Calculate the Curl of this Vector Field
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Calculate the Curl of this Vector Field
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Calculate the Curl of the vector field
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What is an Irrotational field
Irrotational Field: This specifically means that there is no rotational component within the field, which mathematically translates to the curl being zero.
So, if a vector field is irrotational
βΓπΉ=0 (conservative field)
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Define a Definite integral, Line Integral along a curve C and a Closed integral along a curve c
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Compute the Line Integral of F(r) dr
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Show How
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What conditions can be derived from path independence
Path Independence: The integral is independent of the path if:
πΉ= β π where β π is the gradient of π
.
Integration around closed curves in
the Domain always gives 0.
βΓπΉ=0
.
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What's a Systematic Error, how does the error affect the results, how can this error be reduced.
Result from flaws in the measurement process itself, such as instrument calibration, the observerβs bias, or poor measurement techniques.
Error is Consistent and predictable; they can be identified and corrected.
reduced by optimising the experimental technique
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What's a Random Error, how does the error affect the results, how can this error be reduced.
Arise from natural variations in the measurement process, such as environmental fluctuations or inherent limitations in measurement tools.
Error is non-consistent and difficult to identify
Reduced, by repeated measurement and more precise equipment
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What is the Difference between Preciseness and Accuracy
Precise: Indicates the uncertainties are small.
Accurate: Indicates the measurement is close to the known value.
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What are the 3 principle methods for measuring uncertainty
From the measurement scale: Estimate uncertainty as approximately Β±Β½ the smallest division on the measurement tool.
By assessing the technique: Consider observer experience, external influences, published data, and manufacturer's claims.
From the distribution of results: Analyse the spread and variation of repeated measurements using statistical methods.
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What is mean, median and mode
Mean: The average or centroid value, calculated by summing all measurements and dividing by the total number of measurements.
Median: The middle value, where half the measurements are above and half are below. For even-numbered sets, it's the average of the two middle values.
Mode: The value that appears most frequently in the set of measurements.
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What is the formula for a gaussian distribution
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What is a Parent and Sample Distribution
Parent Distribution: Represents the ideal, infinite population with a smooth, continuous curve.
Sample Distribution: Represents a finite, real-world sample with a histogram that approximates the parent distribution.
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Draw a Negative and Positive Skew distribution
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What are the Formulas for Mean, Variance and Standard Deviation.
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Calculate the number of combinations in drawing the following results
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What is the Binomial Distribution formula
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Derive the Poisson Distribution Formula
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Derive the Mean and Standard Distribution of the Poission Distribution
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What is the Formula for Standard Deviation for a Discrete random variable, Binomial Distribution and Sample of Data Points
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What is the Formula for Standard Deviation , and Mean of a Continuous Random Variable
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Transform the equation into the S Domain
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Transform the equation into the S Domain
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Transform the equation into the S Domain
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Transform the equation into the S Domain
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Transform the equation into the S Domain
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Transform the equation into the S Domain, using the power general equation
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Transform the equation into the S Domain
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Transform the equation into the S Domain
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Show the first two derivatives of a polynomial function in the s domain
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Solve the Equation using laplace
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Transfer Equation 4 to the S domain, using the Standard Table
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Transfer Equation 3 to the S domain, using the Standard Table
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Transfer Equation 2 to the S domain, using the Standard Table, and using integration by parts
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Transfer Equation 1 to the S domain, using the Standard Table, and using integration by parts
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Transfer Equation 1 to the t domain, using the Standard Table
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Transfer Equation 2 to the t domain, using the Standard Table
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What is the First Shift Theorem
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Transform the equation into the S Domain
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If a function is delayed by d seconds, show how that changes the expression in the s domain
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What is the Second Shift Theorem
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Use the second shift theorem to find the transform of the function within the s domain
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What are the equations for A0, An and Bn for Fourier series
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What is the equation for fourier series
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What are the Dirichlet Conditions
Such an expression converges to f(t) given the (Dirichlet)
conditions
102
Determine the Fourier expression for the periodic function
Odd function hence A0, An = 0
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Compute the Fourier Series
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For half even wave symmetry what assumptions can you draw on
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For half odd wave symmetry what assumptions can you draw on
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Compute the Fourier Series
107
compute the fourier series of
