(1) Formation:: $y_1 = a sin(wt – {2πx}/{λ})$ & $y_2 = a sin(wt + {2πx}/{λ})$ $y = A sinwt$ … $A = 2a cos {2πx}/{λ}$ (2) $y = 2a cos {2πx}/{λ} sin {2πt}/{T} $ (3) Distance between nodes/antinodes = λ/2 Distance between nodes & antinodes = λ/4 (4) $V = √{{T}/{m}}$ m -> mass per unit length (5) (String) $l = λ/2$ ; $n_o = 1/{2l} √{{T}/{m}} $ (6) Overtones: $n_p = {P+1}/{2}$ (7) Vibration of one end closed air column: l = λ/4 $n_o = 1/{4l} √{{T}/{m}}$ Overtones: $n_p = {2P + 1}/{l}$ (8) Resonance Tube: $n_o = {v}/{4 ...
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(1) $φ = {2πx}/{λ}$ (2) $y = a sin(wt – {2πx}/{λ})$ $y = a sin2π (t/T – x/λ)$ … in +ve x – axis (3) Frequency of beats: $y_1 = asin 2πn_1 t$ $y_2 = asin 2πn_2 t$ y = R sin2πNt … $N = {n_1 + n_2}/{2}$ (4) Max. Intensity: A = ± 2a Beat Frequency: $|n_1 – n_2|$ or $n_1 = n_2 ± \text" Beat frequency"$ (5) Doppler Effect: $n_a = n [{v ± v_o}/{v ± v_s}]$ (6) $φ = {2π(x_2 – x_1)}/{λ}$ (7) Particle velocity: $v = w √{a^2 – y^2}$
(1) $T = F/l (={dw}/{dA})$ (2) When n drops (each of radius r) coalesce into a single drop of radius R, or a single drop of radius R breaks into n drops. $R^3 = n r^3$ Energy released/absorbed = $4 π T = (n r^2 – R^2)$ (3) For a drop, $P_i – P_o = {2T}/{r}$ (4) For a bubble, $P_i – P_o = {4T}/{r}$ (5) 2T cosθ = h r ρ g (6) $T = T_o (1 – αθ)$ θ -> change in temperature (7) $h_1 r_1 = h_2 r_2$
(1) Stress = $F/A$ (2) i. Tensile strain: $E/A$ = ${Mg}/{πr^2}$ ii. Volume = ${A dP}/{A}$ = $dP$ iii. Shearing: $F/A$ (tangential) (3) Strain: ${\text"Change in dimension"}/{\text"Original dimension"}$ (4) i. Longitudinal: $l/L$ ii. Volume: – ${dv}/{V}$ iii. Shearing: tanθ (5) Elastic constants: i. Young’s Modulus: $Y = {FL}/{Al}$ ii. Bulk Modulus: $k = V {dP}/{dV}$ iii. Modulus of Rigidity: $η = F/{A tanθ}$ = ${F}/{Aθ}$ (6) Lateral Strain: $d/D$ (7) Poisson’s Ratio: ${\text"lateral strain"}/{\text"longitudinal strain"}$ = ${dl}/{DL}$ (8) $W = 1/2 \text"load x extension"$ = $1/2 Fl$ (9) Strain Energy/ Unit volume = $U = 1/2 F/A l/L$ i.e. 1/2 ...
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(1) F = -kx (2) $k = m w^2$ (3) ${d^2 x}/{dt^2} + x w^2 = 0$ (4) $PE = 1/2 k x^2$ (5) $TE = 1/2 k a^2$ = $1/2 m w^2 a^2$ = $1/2 m (2πf)^2 a^2$ (6) Composition of SHM $x_{1} = a_{1} sin(wt + α_{1}) $ $x_{2} = a_{2} sin(wt + α_{2})$ x = R sin (wt + d) (7) $R = √{a_1_^2 + a_2_^2 + 2a_1 a_2 cos(α_{1} – α_{2})}$ (8)Simple Pendulum: $θ = x/L$ x -> arc length acceleration = $F/m$ = -${gx}/{L}$ $T = 2π √{{L}/{g}}$ Therefore, $w^2 = g/L$ (9) Resultant amplitude: ...
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(1) $I = Σ m_i r_i_^2$ (system of particles) $I = ∫r^2 dm$ (rigid body) (2) $I = M k^2$ (3) $E_{rot} = 1/2 I w^2$ (Also, subst. w = 2πf) (4) $E_{rolling} = 1/2 M v^2 (1 + {k^2}/{R^2})$ = $1/2 M v^2 (1 + I/{MR^2})$ = $1/2 M w^2 (R^2 + k^2)$ (5) Rolling down an inclined plane, $v = √{{2gh}/{1 + c}}$ & $a = {g sinθ}/{1 + c}$ Here, h = L sinθ & $c = {k^2}/{R^2}$ = ${I}/{MR^2}$ (6) Work done by constant external torque, $∆KE_{rot} = 1/2 I (w_2_^2 – w_1_^2)$ (7) $τ = ...
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(1) $F = G {m_1 m_2}/{r^2}$ G = 6.673 x $10^{-11}$ $m^3$/$kg s^2$ (2) $g = {GM}/{R^2}$ i.e. $weight = {GMm}/{R^2}$ (3) $v_c = √{{GM}/{r}}$ = $√{{GM}/{R + h}}$ = $√{{gR^2}/{R + h}}$ = $√{g_{h} (R+h)}$ Also, $v_{c} = √{gR}$ (4) $T^2 = {4π^2}/{GM} r^3$ i.e. $T^2 α r^3$ (5) Gravitational potential: $V = – {GM}/{r}$ (r≥R) For a satellite, $KE = {GM m }/{2r}$ = ${GMm}/{2(R+h)}$ = $BE$ Total $E = – {GMm}/{2r}$ = – ${GMm}/{2(R+h)}$ (6) For body at rest on Earth’s surface, $v_e = √{{2GM}/{R}}$ = $√{2gR}$ i.e. $v_e = √{2} v_c$ (7) $g_h = g ({R}/{R+h})^2$ ...
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(1) S = rθ (2) $w = θ/t$ (3) $α_{avg.} = {w_2 – w_1}/{t}$ (4) $\ov v$ = $\ov w$ x $\ov r$ (5) a = r α (6) $T = {2π}/{w}$ i.e. $n = {w}/{2π} = {v}/{2π} $ (7) $F = {mv^2}/{r}$ (8) $v_{max} = √{μrg}$ (9) $v_{max} = √{rgtanθ}$ (10) $w = √{{g}/{h}} $ (11) $T = 2π √{{h}/{g}}$ or $T = 2π √{{l cosθ}/{g}} $ i.e. h = l cosθ (12) $v_{top} = √{rg} $ ; $v_{mid} = √{3rg}$ ; $v_{bottom} = √{5rg}$ …. Vertical Circular Motion (13) $T = {mv^2}/{r} + mgcosθ$ (14) $a_{R} = {v^2}/{r}$ ...
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