Task 1

1. Given that the undamped frequency (f) of oscillation of a mass supported on a helical spring depends on the mass (m) and spring stiffness. Using dimensional analysis, we are required to determine which of the equations are not true

The dimensions of the following variables in frequency equations aer:

• Mass, m whose SI unit is kg and Dimension [M]
• Length/Displacement, x whose SI unit is m and Dimension [L]
• Force, F whose SI units is N and dimensional [MLT^-2]
• Frequency, f whose SI units is Hz and dimensional [T^-1]

Equation	Applying Dimensional Analysis	True/False
			False
			True
			False

Therefore, equations a) and c) cannot be true

• Given the standard form of the Bernoulli equation for incompressible non-turbulent fluid flow is:

1. To show that the Bernoulli equation is dimensionally homogeneous

The Pressure P, whose units are, the dimension is:

The units for ρv² are, thus the dimension is:

The units for ρgh are, thus the dimension is:

Therefore, the sum  is dimensionally homogeneous since the units of all parameters are same.

• The dimensions of the constant C is: :

Task 2

1. Given the circular shaft of diameter D rotates at a speed N and is supported by a journal bearing with diametric clearance C. The viscosity of the lubricant is µ and the bearing pressure is p. Then the expression for the friction coefficient f, is expressed as: 

There are five quantities and three dimensions, thus there are 3 dimensionless groups П₁, П₂, and П₃,

For the

• Given that pressure rise P produced by a centrifugal pump depends on the liquid density ρ, the pump rotational speed N, the impeller diameter D and the discharge Q. Deriving the  expression for the relationship between the variables is given below:

There are 4 quantities, thus there are 2 dimensionless groups П₁, П₂,

For Q now:

For a complete equation is:

Task 3

1. Given that a 10mm diameter slotting mill is used in drilling a 75mm thick piece of mild steel. The drill starts each cut from a datum 20mm above the material surface and on the first cut drills 5 mm into the material and then withdraws to the datum.

Total cuts:

Therefore there are 16 cuts in total

The total distance, d moved by the milling cutter

• Given that the gear box of a center lathe is to provide 12 speeds, ranging from 60 RPM to 1800 RPM. If the speeds forms geometric progression, then these speeds are:

We are therefore given the first term and the 12^th term respectively 60 and 1800

Task 4

1. Given the voltage across the capacitor (Vc) is given by the formula:

1. R=1kΩ and C=100μF, then time constant for the circuit is

1. Calculating the capacitor voltage at t=0.25 seconds, and

1. Figure 1 below shows the plot a graph of V_c against time for times between 0s and when the capacitor is ‘fully charged”

Figure 1: Capacitor voltage against time, t

It takes approximately 1 second for the capacitor to be fully charged (at 3 d.p)

• The building site shown in figure 2 below, we are required to determine the total length, P of the wall

Figure 2: The building site

• Given the electric power line of unknown length and mass per unit length of 3.0 kg/m, minimum clearance level =80 m and maximum weight of pylons insulator should be less than 4400 N

Clearance level, c = 80 m

From the figure above, the height H, is given as

The sag, h of the cable is thus:

The distance d, between the pylons:

The length of the cable, L

The maximum tension in the power line

Task 5

Given the data:

1. The mean/Average is given by:

1. The standard deviation is given by

Task 6

1. Given the lengths that are normally distributed with mean 50.00 mm and standard deviation0.24 mm. A pin must have a length between 49.60 mm and 50.35 mm to be considered acceptable
2. The proportion of pins that are scrapped

1. The proportion of pins that require grinding

1. The  proportion of pins that are acceptable

• The two batches of bottle caps are made in HDPE in red and yellow
• The probability that all reds are chosen

• The probability that all yellows are chosen

• The probability that 3 reds and two yellows are chosen

• The probability that 1 yellow and 4 reds are chosen

• 
  

References

Kharagpur, I., 2015. Brief overview of bearings. [Online]
Available at: https://nptel.ac.in/content/storage2/courses/112105125/pdf/mod14les1.pdf