Dimensions and dimensional analysis
- A force defined by F=α t^2+β t acts on a particle at a given time t. The factor which is dimensionless, if α and β are constants, is: Answer: 3. α t/β Key idea: Every term in a physical formula has the same dimensions as F , so α t² and β t have the dimensions of force. Equ... 1
- The quantities which have the same dimensions as those of solid angle are : Answer: 2. strain and angle Key idea: Solid angle (like plane angle) is a ratio of two lengths/areas, so it is dimensionless. Strain (change... 1
- Match List - I with List - II :List - I: (a) Gravitational constant (G) (b) Gravitational potential energy (c) Gravitational potential (d) Gravitational intensityList - II: (i) [L^2T^-2] (ii) [M^-1L^3T^-2] (iii) [LT^-2] (iv) [ML^2T^-2]Choose the corr… Answer: 3. (a) - (ii), (b) - (iv), (c) - (i), (d) - (iii) Find each dimension from its formula. (a) G=Fr²/m² : [M⁻¹L³T⁻²] , so (ii) (b) Pote... 1
- The dimensions [MLT^-2A^-2] belong to the : Answer: 4. magnetic permeability Force has dimension MLT⁻² . Magnetic permeability appears in F=μ₀/4πI₁I₂ l/d , so μ₀ has dimensions of forc... 1
- If E and G respectively denote energy and gravitational constant, then E/G has the dimensions of : Answer: 1. [M²][L⁻¹][T⁰] Key idea: get the dimensions of G from F=Gm₁m₂/r² . [G]=M⁻¹L³T⁻² and [E]=ML²T⁻² . E/G=dfracML²T⁻²M⁻¹L³T⁻²=M²L⁻¹T⁰ .... 1
- If force [F], acceleration [A] and time [T] are chosen as the fundamental physical quantities. Find the dimensions of energy. Answer: 2. [F][A][ T² ] Key idea: write energy as work = force × distance, and express the distance through acceleration and time. L = 1/2 A... 1
- Dimensions of stress are : Answer: 4. [ML⁻¹T⁻²] Stress =force/area . [stress]=frac[MLT⁻²][L²]=[ML⁻¹T⁻²] This is the same as the dimensions of pressure. Answer: option... 1
- A physical quantity of the dimensions of length that can be formed out of c, G and e^2/4πε_0 is [c is velocity of light, G is universal constant of gravitation and e is charge] : Answer: 4. 1/c²[Ge²/4πε₀]^(1/2) Key idea: match dimensions. e²/4πε₀ is force × length ² : [ML³T⁻²] . G=[M⁻¹L³T⁻²] . Product Ge²/4πε₀=[L⁶T⁻⁴]... 1
- Planck's constant (h), speed of light in vacuum (c) and Newton's gravitational constant (G) are three fundamental constants. Which of the following combinations of these has the dimension of length? Answer: 4. frac√(hG)c^(3/2) Write the dimensions: [h]=ML²T⁻¹ , [c]=LT⁻¹ , [G]=M⁻¹L³T⁻² . Length has the dimension of √(Gh/c³) (the Planck le... 1