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Dynamical quantities associated with SHM | Oscillations and waves

Dynamical quantities associated with SHM Displacement The displacement of a particle executing simple harmonic motion is Velocity Acceleration Phase The angle of different physical quantities (displacement, velocity and acceleration) of S.H.M. are called its phase. If t = 0, then ቀ 0 = θ (initial phase or epoch) Time period The time taken by particle to complete one oscillation is known as time period. Frequency The reciprocal of time period is known as frequency (ν). ∴ ν = 1/T = 1/2π√(k/m) To know more about dynamical quantities associated with simple harmonic motion this lecture please click on the link for English and click on the link for Hindi https://monetag.com/?ref_id=nU6C Our other websites https://www.sacademy.co.in https://blog.sacademy.co.in https://bhakti.sacademy.co.in http://food.sacademy.co.in

Postulates of classical statistical mechanics | Statistical mechanics

Postulates of classical statistical mechanics It is based on the following postulates Postulates of equal a priori probability According to this postulate the probability of finding the phase point for given system in any region of the Γ-space is identical with that for any other region of equal extension or volume. Or For a system in equilibrium, all accessible microstates corresponding to a given macro-state are equally probable. Thus in thermodynamic equilibrium the system under consideration is a member of an ensemble (micro-canonical ensemble) with a density function ρ (p, q) and the value of density function is given according to the following rule. If E < H (p, q) < E + ΔE, ρ (p, q) = constant, otherwise ρ (p, q) = 0 Thus all members of the ensemble have the same number of particles N and same volume V. Let f (p, q) represents any measurable property of the system. If the postulate of equal a priori probability is useful, then the average v...

Phase space and density function | Statistical mechanics

Phase space and density function Phase space or 𝚪 space In classical mechanics the position of a point particles is described in terms of three Cartesian coordinates x, y, z. And the state of motion of particle is described in terms of velocity component ẋ, ẏ, ż or momentum coordinates p x , p y , p z . We imagine a 6-d space in which the six coordinates are x, y, z and p x , p y , p z are marked along six mutually perpendicular axes in space. The combined position and momentum space is known as phase space or Γ space . A point in the phase space represents the position and momentum of the particle at some particular instant. Density function Let a classical system has a large number of molecules (N) occupying a large volume V. Generally N = 10 23 molecules and V = 10 23 molecular volumes or N → ∞ and V → ∞ N/V = v ; here v = a specific volume, which is a finite number. The system will be regarded as isolated in the sense that the ener...

Moseley law in Hindi | मोज़ले का नियम

मोज़ले का नियम इस नियमानुसार एक्स किरण स्पेक्ट्रम में किसी स्पेक्ट्रमी रेखा की आवृत्ति (frequency), जिस तत्व से एक्स किरण उत्सर्जित होती है, उसके परमाणु क्रमांक (atomic number) के वर्ग के समानुपाती होती है। यदि एक्स किरण स्पेक्ट्रम में स्पेक्ट्रमी रेखा की आवृत्ति (𝜈) तथा जिस तत्व से एक्स किरण उत्सर्जित होती है उसका परमाणु क्रमांक (Z) हो, तो 𝜈 = Z 2 ⇒ √𝜈 ∝ Z या √𝜈 = a (Z – b ) यहां (Z – b ) एक्स किरण उत्सर्जित करने वाले तत्व का प्रभावी परमाणु क्रमांक तथा b अभिलाक्षणिक नियतांक या आवरणांक है। मोज़ले नियम की उपयोगिता किसी तत्व के अभिलाक्षणिक गुण उसके परमाणु क्रमांक पर निर्भर करते हैं, द्रव्यमान पर नहीं। इसलिए आवर्त सारणी में कुछ तत्वों की स्थिति पुर्नव्यवस्थित की गई। आवर्त सारणी के कुछ नए तत्वों की खोज की गई। जैसे Hf (72), In (61), Re (75) आदि। बोहर सिद्धान्त के आधार पर मोज़ले नियम की व्याख्या n 1 तथा n 2 कोश में इलेक्ट्राॅन की ऊर्जा चूंकि धनात्मक नाभिक अन्य इलेक्ट्राॅनों द्वारा ढक दिया जाता है। इसलिए ह...

Moseley’s law | Atomic and Molecular physics

Moseley’s law According to it, the frequency of a spectral line in X-ray spectrum varies as the square of the atomic number of the element emitting it. If 𝜈 is the frequency of spectral line in X-ray spectra and Z is the atomic number of element, then 𝜈 = Z 2 ⇒ √𝜈 ∝ Z or √𝜈 = a (Z – b ) Here (Z – b ) is the effective atomic number of emitting element and b is the characteristic constant or screening constant. Importance of Moseley’s law The characteristic property of an element depends on their atomic number, not on their masses. So the position of some elements in periodic table was rearranged. On the basis of Moseley law some new elements of periodic table was discovered like Hf (72), In (61), Re (75) etc. Explanation of Moseley’s law on the basis of Bohr’s theory The energy of electron in n 1 and n 2 orbits are given by Since the positive nucleus is screened by rest of the electrons. Therefore we...

Invariance of Poisson bracket under canonical transformation | Classical Mechanics

Invariance of Poisson bracket under canonical transformation Let u and v be two functions such that u = u (q i , p i , t) and v = v (q i , p i , t) Let a canonical transformation is from (q i , p i , t) → (Q i , P i , t) Here q = q (Q, P, t) and p = p (Q, P, t) Corresponding to it the transformation in u and v are u (q i , p i , t) → u′ (Q i , P i , t) and v (q i , p i , t) → v′ (Q i , P i , t) Now we have to prove that if (q, p, t) → (Q, P, t) is canonical then [u, v] p, q = [u′, v′] P, Q It means the Poisson bracket are invariant under a canonical transformation. Proof If F 1 and F 2 are generating function, then the transformation relation for the variables are F 2 = F 1 + PQ Thus the Poisson brackets are invariant under a canonical transformation . To know more about Invariance of Poisson bracket under canonical transformation click on the link for English and click on the link for Hindi

Angular momentum Poisson brackets | Classical Mechanics

Angular momentum Poisson brackets Angular momentum involving Poisson bracket If r is position vector, and p is linear momentum, then angular momentum L = r × p Thus the Poisson bracket between any pair of the components satisfy [L i , L j ] = ε ijk x j p k Proof To know more about Angular momentum Poisson bracket click on the link for English and click on the link for Hindi

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