{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

14%20-%20Oscillations - 14 OSCILLATIONS Page 1 14.1...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
14 - OSCILLATIONS Page 1 14.1 Periodic and Oscillatory motion Motion of a system at regular interval of time on a definite path about a definite point is known as a periodic motion, e.g., uniform circular motion of a particle. To and fro motion of a system on a linear path is called an oscillatory motion, e.g., motion of the bob of a simple pendulum. 14.2 Simple harmonic motion This is the simplest type of periodic motion which can be understood by considering the following example. Suppose a body of mass m is suspended at the lower end of a massless elastic spring obeying Hooke’s law which is fixed to a rigid support in the vertical position. The spring elongates by length Δ l and attains equilibrium as shown in Fig. ( b ) Here two forces act on the body. ( 1 ) Its weight, mg, downwards and ( 2 ) the restoring force developed in the spring, k Δ l , upwards, where k = force constant of the spring. For equilibrium, mg = k Δ l ( 1 ) The spring is constrained to move in the vertical direction only. Now, suppose the body is given some energy in its equilibrium condition and it undergoes displacement y in the upward direction as shown in Fig. ( c ). Two forces act on the body in this displaced condition also. ( 1 ) Its weight, mg, downwards and ( 2 ) the restoring force developed in the spring, k ( Δ l - y ), upwards. The resultant force acting on the body in this condition is given by F = - mg + k ( Δ l - y ) ( 2 ) From equations ( 1 ) and ( 2 ), F = - ky
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
14 - OSCILLATIONS Page 2 Displacement: The distance of the body at any instant from the equilibrium point is known as its displacement at that instant. The displacements along the positive Y-axis are taken as positive and those on the negative Y-axis are taken as negative. In the equation, F = - ky, F is negative when y is positive and vice versa. Thus, the resultant force acting on the body is proportional to the displacement and is directed opposite to the displacement, i.e., towards the equilibrium point. Differential equation of simple harmonic motion ( SHM ) According to Newton’s second law of motion, F = ma = m dt dv = m 2 2 dt y d = - ky ( for spring-type oscillator as above ) 2 2 dt y d = - m k y = - ω 0 2 y ( taking m k = ω 0 2 ) 2 2 dt y d + ω 0 2 y = 0 This is the differential equation of SHM. To obtain the solution of the above differential equation is to obtain y as a function of t such that on twice differentiating y w.r.t. t, we get back the same function y with a negative sign. Both the sine and the cosine functions possess such a property. Hence, taking y = A 1 sin ω 0 t + A 2 cos ω 0 t as a possible solution and differentiating twice w.r.t. t, dt dy = A 1 ω 0 cos ω 0 t - A 2 ω 0 sin ω 0 t and 2 2 dt y d = - A 1 ω 0 2 sin ω 0 t - A 2 ω 0 2 cos ω 0 t = - ω 0 2 ( A 1 sin ω 0 t + A 2 cos ω 0 t ) = - ω 0 2 y Thus, y t = A 1 sin ω 0 t + A 2 cos ω 0 t is the solution of the differential equation and is known as its general solution, where y t is the displacement of simple harmonic oscillator ( SHO ) at time t.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern