Harmonic motion11/30/2023 ![]() ![]() Note that the initial condition determines both φ and A. Here we start with an initial velocity, which is In the second movie shown at right, however, the mass is given an impulsive start, so the initial condition approximates maximum velocity and x = 0 at t = 0. (Indeed, for this particular case, we could say that the curve is a cos function rather than a sine.) So the amplitude is maximal (x = A) at t = 0, so the required phase constant is φ = π/2. In the first movie shown shown at right, the mass is released from rest, Where A is the amplitude, and the phase constant φ is determined by the initial conditions. However, we can verify by subsitution that the solution is Solving this particular equation is described in detail on the Differential Equations page. Newton's second law states that the acceleration d 2x/dt 2 of the mass m subject to total force F satisfies F = m.d 2x/dt 2 , which gives the differential equation There is also a page on the Kinematics of Simple Harmonic Motion. ![]() However, we do a quantitative analysis on the multimedia chapter Oscillations and also solve this problem as an example on Differential Equations. The analysis that follows here is fairly brief. In this case, k = k 1 + k 2, where k 1 and k 2 are the constants of the two springs. The force F exerted by the two springs is F = − kx, where k is the combined spring constant for the two springs (see Young's modulus, Hooke's law and material properties). The cycle then repeats exactly, so the motion is periodic.įor linear springs, this leads to Simple Harmonic Motion. Consequently, the system returns to its initial condition. The spring force now acts to the left, so it decelerates until it stops at its maximum rightwards displacement.īecause no non-negligible nonconservative forces act, mechanical energy is conserved.
0 Comments
Leave a Reply.AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |