Ch08 Periodic Motion

PHYSICS 1ST PAPER CHAPTER 8 Periodic Motion Question 1 d2𝑥 d𝑡2 + 𝜔2𝑥= 0 is a differential equation of a simple harmonic motion. When asked to write its solution, Shamim wrote 𝑥= 𝐵cos(𝜔𝑡+ 𝛼) and Rimjhim wrote 𝑥= 𝐶tan(𝜔𝑡+ 𝜑). (a) What is called periodic motion? (b) What do you mean by temporal periodicity? (c) Show that 𝑥= 𝐴sin(𝜔𝑡+ 𝛿) is a solution of the differential equation of simple harmonic motion. (d) Verify by mathematical analysis whether the solutions of Shamim and Rimjhim are proper or not. Solution (a) A motion that repeats itself after equal intervals of time is called periodic motion. (b) Some events repeat after a fixed interval of time. This repetition with time is called temporal periodicity. The fixed interval is the time period 𝑇. The state of the body at a time 𝑡returns at the time 𝑡+ 𝑇. The oscillation of a pendulum is an example. The rotation of the earth is another example. (c) The given equation is 𝑥= 𝐴sin(𝜔𝑡+ 𝛿) Differentiating with respect to time 𝑡, d𝑥 d𝑡= 𝐴𝜔cos(𝜔𝑡+ 𝛿) Differentiating once more, d2𝑥 d𝑡2 = −𝐴𝜔2 sin(𝜔𝑡+ 𝛿) = −𝜔2𝑥 d2𝑥 d𝑡2 + 𝜔2𝑥= 0 The equation satisfies the differential equation. Hence it is a solution of simple harmonic motion. (d) A proper so

Solution of Shamim. Here 𝑥= 𝐵cos(𝜔𝑡+ 𝛼). d𝑥 d𝑡= −𝐵𝜔sin(𝜔𝑡+ 𝛼) d2𝑥 d𝑡2 = −𝐵𝜔2 cos(𝜔𝑡+ 𝛼) = −𝜔2𝑥 d2𝑥 d𝑡2 + 𝜔2𝑥= 0 This solution satisfies the differential equation. Solution of Rimjhim. Here 𝑥= 𝐶tan(𝜔𝑡+ 𝜑). d𝑥 d𝑡= 𝐶𝜔sec2(𝜔𝑡+ 𝜑) d2𝑥 d𝑡2 = 2𝐶𝜔2 sec2(𝜔𝑡+ 𝜑) tan(𝜔𝑡+ 𝜑) = 2𝜔2𝑥sec2(𝜔𝑡+ 𝜑) The right side is not equal to −𝜔2𝑥. This solution does not satisfy the differential equation. 𝜋 2𝜋 3𝜋 4𝜋 −𝐵 𝐵 cosine tangent 𝜔𝑡 𝑥 The cosine curve stays between −𝐵and +𝐵. The tangent curve becomes infinite again and again. A real oscillating body cannot have an infinite displacement. The solution of Shamim is proper. The solution of Rimjhim is not proper. ahbab.dev 2

Question 2 A particle is revolving around the point 𝑂in a circular path 𝐴𝐵𝐶𝐷of radius 0.02 m along the clockwise direction with an angular velocity 1 rad s−1 and in 10 s it comes to 𝑃from 𝐴. 𝑋 𝑌 𝐴 𝐵 𝐶 𝐷 𝑂 𝑃 𝑁 𝜃 𝜃 𝜔 𝑥 (a) What is called simple harmonic motion? (b) Explain the relation of force constant and mass in case of a spring. (c) Deduce the differential equation of the motion along the 𝑋-axis of the particle of the stimulus. (d) If the initial phase of the particle is 0, then what is the displacement at 𝑃along the 𝑋-axis? Solution (a) In simple harmonic motion, the acceleration is proportional to the displacement. It is always directed towards the equilibrium position. (b) Let a body of mass 𝑚oscillate on a spring of force constant 𝑘. 𝜔= √︂ 𝑘 𝑚, 𝑇= 2𝜋 √︂𝑚 𝑘 A larger force constant means a stiffer spring. The restoring force is then greater and the time period is smaller. A larger mass has more inertia. The time period is then greater. For a fixed time period, 𝑘is proportional to 𝑚. (c) Let the radius of the circle be 𝑅. The particle starts from 𝐴and moves clockwise. In a time 𝑡it turns through an angle 𝜃= 𝜔𝑡from 𝑂𝐴. The foot of the perpendicular from 𝑃on the 𝑋-axis is 𝑁. In th