Find dimensions of compound quantity

A question is this type if and only if it asks to find the dimensions of a quantity defined as a ratio, product, or combination of other quantities (e.g., Young's modulus, viscosity, surface tension).

2 questions · Moderate -0.4

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OCR MEI Further Mechanics A AS 2021 November Q1
4 marks Easy -1.2
1 The specific energy of a substance has SI unit \(\mathrm { J } \mathrm { kg } ^ { - 1 }\) (joule per kilogram).
  1. Determine the dimensions of specific energy. A particular brand of protein powder contains approximately 345 Calories (Cal) per 4 ounce (oz) serving. An athlete is recommended to take 40 grams of the powder each day. You are given that \(1 \mathrm { oz } = 28.35\) grams and \(1 \mathrm { Cal } = 4184 \mathrm {~J}\).
  2. Determine, in joules, the amount of energy in the athlete's recommended daily serving of the protein powder.
OCR Further Mechanics 2023 June Q2
11 marks Standard +0.3
Materials have a measurable property known as the Young's Modulus, \(E\). If a force is applied to one face of a block of the material then the material is stretched by a distance called the extension. Young's modulus is defined as the ratio \(\frac{\text{Stress}}{\text{Strain}}\) where Stress is defined as the force per unit area and Strain is the ratio of the extension of the block to the length of the block.
  1. Show that Strain is a dimensionless quantity. [1]
  2. By considering the dimensions of both Stress and Strain determine the dimensions of \(E\). [2]
It is suggested that the speed of sound in a material, \(c\), depends only upon the value of Young's modulus for the material, \(E\), the volume of the material, \(V\), and the density (or mass per unit volume) of the material, \(\rho\).
  1. Use dimensional analysis to suggest a formula for \(c\) in terms of \(E\), \(V\) and \(\rho\). [5]
  2. The speed of sound in a certain material is \(500\) m s\(^{-1}\).
    1. Use your formula from part (c) to predict the speed of sound in the material if the value of Young's modulus is doubled but all other conditions are unchanged. [1]
    2. With reference to your formula from part (c), comment on the effect on the speed of sound in the material if the volume is doubled but all other conditions are unchanged. [1]
  3. Suggest one possible limitation caused by using dimensional analysis to set up the model in part (c). [1]