Answer:
Explanation:
a) the steady-state, 1-D incompressible and no energy generation equation can be expressed as follows:

b) For a transient, 1-D, constant with energy generation
suppose T = f(x)
Then; the equation can be expressed as:

where;
= heat generated per unit volume
= Thermal diffusivity
c) The heat equation for a cylinder steady-state with 2-D constant and no compressible energy generation is:

where;
The radial directional term =
and the axial directional term is 
d) The heat equation for a wire going through a furnace is:
![\dfrac{\partial ^2 T}{\partial z^2} = \dfrac{1}{\alpha}\Big [\dfrac{\partial ^2 T}{\partial ^2 t}+ V_z \dfrac{\partial ^2T}{\partial ^2z} \Big ]](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20%5E2%20T%7D%7B%5Cpartial%20z%5E2%7D%20%3D%20%5Cdfrac%7B1%7D%7B%5Calpha%7D%5CBig%20%5B%5Cdfrac%7B%5Cpartial%20%5E2%20T%7D%7B%5Cpartial%20%5E2%20t%7D%2B%20V_z%20%5Cdfrac%7B%5Cpartial%20%5E2T%7D%7B%5Cpartial%20%5E2z%7D%20%5CBig%20%5D)
since;
the steady-state is zero, Then:
'
e) The heat equation for a sphere that is transient, 1-D, and incompressible with energy generation is:

Answer:
A)The sketches for the required planes were drawn in the first attachment.
B)The sketches for the required directions were drawn in the second attachment.
To draw a plane in a simple cubic lattice, you have to follow these instructions:
1- the cube has 3 main directions called "a", "b" and "c" (as shown in the first attachment)
2- The coordinates of that plane are written as: π:(1/a₀ 1/b₀ 1/c₀) (if one of the coordinates is 0, for example (1 1 0), c₀ is ∞, therefore that plane never cross the direction c).
3- Identify the points a₀, b₀, and c₀ at the plane that crosses this main directions and point them in the cubic cell.
4- Join the points.
To draw a direction in a simple cubic lattice, you have to follow these instructions:
1- Identify the points a₀, b₀, and c₀ in the cubic cell.
2- Draw the direction as a vector-like (a₀ b₀ c₀).
Answer:
Logarithmic decrement is equal to 0.182
Explanation:
given,
amplitude decay = 9 dB
number of cycles = 12 cycles
mass of the system = 7 kg
spring stiffness = 3000 N/m
logarithmic decrement = ?
now,
logarithmic decreament = 
= 
=ln (1.2)
= 0.182
Hence, Logarithmic decrement is equal to 0.182