Plane angle and solid angle have

Plane angle and solid angle have: 1.both units and dimensions2.units but no dimensions3.dimensions but no units4. no units and no dimensions Units and Measurement Physics NEET Practice Questions, MCQs, Past Year Questions (PYQs), NCERT Questions, Question Bank, Class 11 and Class 12 Questions, and PDF solved with answers • Recommended MCQs - 76 Questions • Recommended PYQs (STRICTLY NCERT Based) • Mini Q Banl-Units and Measurement • NCERT Solved Examples Based MCQs • NCERT Exercise Based MCQs • NCERT Exemplar (Objective) Based MCQs • AR & Other Type MCQs • Padma Shri H C Verma (Objective Exercises) Based MCQs • Past Year (2019 onward - NTA Papers) MCQs • Past Year (2016 - 2018) MCQs • Past Year (2006 - 2015) MCQs • Past Year (1998 - 2005) MCQs Match List-I with List-II: List-I List-II (a) Gravitational constant(G) (i) [L 2T -2] (b) Gravitational potential energy (ii) [M -1L 3T -2] (c) Gravitational potential (iii) [LT -2] (d) Gravitational intensity (iv) [ML 2T -2] Choose the correct answer from the options given below: (a) (b) (c) (d) 1. (iv) (ii) (i) (iii) 2. (ii) (i) (iv) (iii) 3. (ii) (iv) (i) (iii) 4. (ii) (iv) (iii) (i) The determination of the value of acceleration due to gravity \((g)\) by simple pendulum method employs the formula, \(g=4\pi^2\frac\Big)\) Consider a simple pendulum, having a bob attached to a string, that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length ( l ), t...

For a project, I'm working on calculating solid angles using calculus, but when I test my formula for the solid angle of a regular tetrahedron, I end up with $0.4633$ when according to Wolfram, I should get $0.5513$. This is going to be a bit long... The solid angle is equal to the spanned area ($\Omega$) on the unit sphere. We also know that the area spanned is directly proportional to the volume spanned ($V$) by the solid angle. $$\frac(\sqrt3/3)$. However, my equation returns $0.4633$ steradians. I have no idea where I went wrong. It's probably a bit counter-intuitive but a triangle in $(\theta,\phi)$ does not map to a spherical triangle, except for $\theta=0$ line (equator) or $\phi=\rm const$ (meridian). The straight line maps to a strange curve on the sphere. For instance, a simple line at $\theta=\rm const$ (except for equator) is "bulged" out and isn't the shortest path between two points on a sphere. Integration in spherical coordinates is not the best way to do this, if you have a polygon on a sphere. This is because spherical coordinates have a preferred direction along $z$ and don't preserve straight lines. Moreover, you did a long detour through volume, even though the spherical coordinates alredy remove the radius from the equation (which you figured out by yourself). What you need is spherical geometry that is meant precisely for this kind of calculations: Using calculus, we can get a generalized formula (derived in For a regular tetrahedron, n=no. of edges ...

I have learnt that the dimensionless quantities have no unit. Whoever you learnt this from is incorrect . Clearly, dimensionless quantities CAN have units, as you have figured out in the case of angles. Another example of dimensionless quantity with units is the relative abundance of particles which has units of ppm(parts per million) , ppb(parts per billion) etc. From 7.10 Values of quantities expressed simply as numbers: the unit one, symbol 1\ Certain quantities, such as refractive index, relative permeability, and mass fraction, are defined as the ratio of two mutually comparable quantities and thus are of dimension one (see Sec. 7.14). The coherent SI unit for such a quantity is the ratio of two identical SI units and may be expressed by the number 1. However, the number 1 generally does not appear in the expression for the value of a quantity of dimension one. For example, the value of the refractive index of a given medium is expressed as $n = 1.51 \times 1 = 1.51$. On the other hand, certain quantities of dimension one have units with special names and symbols which can be used or not depending on the circumstances. Plane angle and solid angle, for which the SI units are the radian (rad) and steradian (sr), respectively, are examples of such quantities... The rationale beyond units for dimensionless quantities is to keep memory of the quantity used as the denominator of the ratio. Thanks for contributing an answer to Physics Stack Exchange! • Please be sure to answ...

Let , and be the edges of a trihedron that determines a solid angle. The plane angles opposite the edges are denoted , , , and the angles between the edges and their opposite faces are denoted , , . Construct three planes parallel to the faces , and at distance 1 from the corresponding faces. Let the intercepts of these planes with edges of the solid angle be , , . Also define the points , , , such that , , , , to get a parallelepiped with all faces of equal area, since all heights are equal. The lengths of the edges are , and . The areas of the faces are , and . • Supplementary Solid Angles for Trihedron • A Theorem on the Dihedral Angles of a Tetrahedron • Right-Angled Tetrahedron • Relations between the Platonic and Archimedean Solids • Construct a Dihedral Angle of a Tetrahedron Given Its Plane Angles at a Vertex • Spherical Cosine Rule for Angles • Three Planes in a Trihedron • Locus of Points at Constant Distance from the Edges of a Trihedron • Theorem on the Dihedral Angles of an Isosceles Tetrahedron • Dissecting the Triangular Hebesphenorotunda (Johnson Solid 92)