Dimensional Analysis, Examination Paper - Physics - Prof IB Leader, Exams of Physics

Download Dimensional Analysis, Examination Paper - Physics - Prof IB Leader and more Exams Physics in PDF only on Docsity! Engineering FIRST YEAR Part IA: Dimensional Analysis EXAMPLES PAPER 1 1 Voluntary/optional “warm-up” questions are marked with a V. Straightforward questions are marked with a †, Tripos standard with a *. V1. Using the unit conversion factors given in lectures and in the appendices of the “Guide to Units”, complete the following: (a) 6 ft = .............................. m (b) 2 kg = .............................. lb (c) 70 mph = .............................. m/s (speed) †2. Using the unit conversion factors given in lectures and in the appendices of the “Guide to Units”, complete the following: (a) 1 year = .............................. s (b) 1 ft3/min = .............................. m3/hour (volumetric flow rate) (c) 1 kg/m3 = .............................. lb/ft3 (density) (d) 1 lbf/in2 = .............................. N/m2 (pressure or stress) The lbf (pound force) is defined in Appendix D of the “Guide to Units”. V3. Using the definitions given in the “Guide to Units”, find the dimensions (in the M-L-T-Θ system) of the following quantities: (a) Energy (b) Power (c) Thermal conductivity †4. Which of the following equations appear to be dimensionally inconsistent? In other words, in which equations do the constants have dimensions? (a) The formula used by heating contractors to determine the heating requirements of a room: Q = 0.04V + W + 0.33A Q = heat supply for room per °F temperature difference between inside and outside expressed in Btu/hour °F (a British Thermal Unit, Btu, is a measure of energy), V = volume of room expressed in ft3, W = area of windows expressed in ft2, A = area of external walls expressed in ft2. (b) The ‘White’ formula for the tension left in a straight weld joining two steel plates, on account of the shrinkage of the weld-metal: T = 0.2 v Q T = tension, Q = electrical power input to the welding arc, v = velocity of welding arc along the weld-line. [Continued overleaf] Engineering Part IA Dimensional Analysis Examples Paper 1 2 (c) The ‘Chezy’ formula for the mean flow velocity of water in a sloping pipe, the cross- section of which is not necessarily circular: u = C L AS u = mean flow velocity, A = cross-sectional area of pipe, S = slope of pipe, L = ‘wetted perimeter’ of cross-section, C = constant. V5. You are considering a dimensional analysis problem that has N variables containing M dimensions. What is the minimum number of dimensionless groups you can form if (a) N = 3, M = 2 (b) N = 5, M = 3 (c) N = 3, M = 3? V6. Find as many dimensionless groups (that are independent of each other) as you can from the following sets of variables: (a) Power P, mass m, speed V, length L (b) Pressure p, density ρ, speed V, gravitational acceleration g, height h 7. When a circular disc of material is rotating about a central axis that is perpendicular to the plane of the disc, it will ‘burst’ under the effects of its own inertia loading if a critical angular velocity ωc (rad/s) is exceeded. In tests carried out on gas turbine discs, it is found that the value of ωc is dependent only on the maximum stress σ (N/m2) that the material can withstand, the density ρ (kg/m3) of the material and the radius R of the disc. What is the form of the relationship governing ωc, ρ, σ and R? Use the elimination method to perform any dimensional analysis required. 8. A construction company is required to build a long road bridge across a marsh. There are many short spans but there is to be one long span in the middle. The figure below shows a stage of construction when all the short spans have been completed and the two arms of the long span are being extended so that they meet at the centre. The ground conditions are such that no supporting structure can be built. The chief engineer is concerned that the individual arms of the incomplete long span may collapse under their own weight before they can be joined together, in the manner indicated by the broken lines above. The bridge will be constructed using girders made from steel. The steel has a density ρ of 7843 kg/m3 and can withstand a maximum stress σ of 400 MN/m2. A simple model of one half of the long span is constructed out of aluminium. The half-span is 500 mm. The aluminium has a density ρ of 2720 kg/m3 and can withstand a maximum stress σ of 70 MN/m2. Using a centrifuge, it is found that the scale model collapses when the acceleration reaches 400 m/s2. (The use of the centrifuge allows the acceleration due to gravity g applied to the model to be varied.) What is the half-span of the largest steel bridge that can be constructed in this way using this particular design? Use the elimination method to perform any dimensional analysis required.