Time Period of Periodic and Simple Harmonic Motion (SHM)

 1 SHM F=-kx or ma=-kx or a=-(k/m)x =-ω2 x and T= 2π /ω =  2π√(k/m). We find m/k=x/a. Hence T=2π√(displacement/acceleration) 2 Combination of springs For Series -- 1/k = 1/k1  + 1/k2        For parallel , k=k1+k2 3. Simple pendulum with small angle of oscillation sothat sinθ ≃ θ mgcosθ =T, mgsinθ =-ma or gθ=-a or a=-gθ =-gx/l and T = 2π /ω =  2π√(l/g) 4. Torsion Pendulum hung with metallic wire of length l, having mass m, and radius r and moment of Inertia I = (1/2)mr2 , angle of twist θ Torque=τ =Iα =-cθ where c is the moment of force. α =-(c/I)θ , T= 2π /ω =  2π√(I/c) .Since c=F x 2r =2Fr, T=π√(mr/F) when r --> 0, F remaining constant, T --> 0 i.e. time speeds up. When r=0, there is no oscillation  since it becomes a point mass. A point mass has no dimension since it does not occupy any space  but at the same time , it has a spatial signature in the form of  coordinates. Compare this with energy which has neither spatial signature nor dimension. This dichotomy in the treatment of point mass is reflected in quantum mechanical treatment where we refer to eigen states for both matter and energy. 5. A mass m hung from ceiling by rope of length l moves making an angle θ in a circle of radius r on the table. What is T? Tcosθ =mg      , sinθ =r/lTsinθ=mv2/r  , time period t=2πr/v. Solving, we get t= 2π√(lcosθ/g) 6. A mass m hung from ceiling of a horizontally moving vehicle with acceleration a by rope of length l T= 2π√(l/a') where a'2 =g2+a2 ; 7. A mass M is hung from the hole in a table and a string of radius r with hole at the origin with mass m at the end executes a circle so that the M does not fall down mv2/r=Mg or mω2 r =Mg , ω =√(Mg/mr) , T=2π√(mr/Mg) 8. A body of mass m executes a SHM in a straight line. If maximum velocity is a and maximum acceleration is b, then displacement =y = a'sinωt, velocity=v=a'ωcosωt , acceleration=a= -a'ω2sinωtmaximum velocity=a=a'ω  . maximum acceleration=b=a'ω2  . a/b=1/ω , T=2π /ω =2π a/b 9. A body of mass m executes SHM. The displacement and velocity at 2 different points are v1,v2 and x1 and x2 respectively. A is amplitude v12 = ω2 (A2-x12) ,  v22 = ω2 (A2-x22). Solvong, we get   ω2=(v12-v22) /(x22-x12),  T=2π /ω =2π √[(x22-x12)/(v12-v22) ] 10. In uniform circular motion, if radius is a , speed is v(If speed will not be constant, then the acceleration will not be towards the center.) F=GMm/a2 =mv2/r  and T=2πa /v and T2 =4π2a2 /v2     .So T2=(4π2/GM)*a3; In motion in an ellipse under the influence of a central force where the heavier mass M is at focus, and semi major axis is a T2=(4π2/GM)*a3 11 If a hole is drilled across the earth passing through the center and a mass of m is dropped into it, it will execute SHM. Let the particle be at a distance x from center where x < R. F=-GMm/x2 =-G(4/3)πx3dm/x2 =maa=-G(4/3)πxd , ω2=4Gπd/3 and T=2π√(3/4Gπd) 12 A sphere of mass m carrying positive charge q is hung from ceilingby a rope of length l amid a uniform electric field E directed vertically upward. F=mg-qE=ma  So a= g-(q/m)E  and T=2π√(l/a) 13. A point mass m is suspended at the end of a massless wire of length l and cross section A. If Y is Young's modulus of elasticity for the wire, rime period of oscillation along vertical line is T Y =(F/A) / (Δl/l) =Fl/(AΔl) or F=YAΔl /l =-ma and so a=-(YA/ml)Δl , T= 2π /ω =  2π√(ml/YA) 14 A weighted glass tube is floating in a liquid with l of its length ommersed. It is pushed down some distance x and released. It executes a SHM. upward thrust=Aldg , At equilibrium, mg=Aldg or m=AldF=Adg(l+x) -Adgl =adgx =-ma or a =-Adgx/Ald =-gx/l  , T = 2π /ω =  2π√(l/g) 15. A vertical U tube of uniform area of cross section a has water up to a height l. If water on one side is depressed and released, the motion is SHM. m=A2ldg , F=-Ald2x =ma or a=-gx/l  , T = 2π /ω =  2π√(l/g) 16. The transverse vibration of a stretched string  of length l tied to 2 ends creates a SHM. T=(l/k)√(m/t) where m is mass per unit length, k is a number which for fundamental frequency is 1/2 and t is tension along the string. k=1/2, 1, 3/2, 2 etc 17. The transverse vibration of a both side open organ pipe of length l  creates a SHM. T=(l/k)√(d/γp) where d is density , p is pressure and γ =Cp / Cv and k=1/2, 1, 3/2, 2 etc The transverse vibration of a one side open organ pipe of length l  creates a SHM.(At closed end, node is formed) T=(l/k)√(d/γp)  and k=1/4, 3/4, 5/4, 7/4 etc 18. Gas is enclosed in a container with area of cross section A, pressure P and volume V with a piston on the top.Piston is slightly pushed downward by x and released and it executes a SHM. PVγ=constant (P+dP)(V-dV)γ =constant or PVγ(1+dP/P)(1-dV/V)γ =constant  or (1+dP/P)(1-dV/V)γ =1Expanding the bold terms bionomially up to 2nd term, (1+dP/P)(1-γdV/V) =1 or dP/P=γdV/V or dP=γPdV/V F=AP =γPAdV/V =γPAAx/V=γPA2x/V =-ma o a =-x( γPA2/mV) and T= 2π /ω =  2π√(mV/γPA2). 19 In an LC circuit in oscillation of charge and current, T =2π√(LC) 20 A pendulum of mass m with length l is suspended in a car that is travelling with a constant speed u around a circle of radius r. If the pendulum undergoes small oscillations, time period T is T =T =2π√(l/a) where a2 =g2 -a'2  where a' (relative acceleration)=v2/r and a'2 =v4 /r 2 21 For planetary motion around sun in elliptical orbits T =2π√(a3/GM)