11/27/2023 0 Comments Harmonic motion![]() If the particle is pulled aside and released, it oscillates in a circular arc with the center at the point of suspension ‘O’.Īt angle θ, the forces acting on the particle are:Īs the particle makes pure rotation about a horizontal axis through O (say OA), let us find the torques of the forces acting on it. This is called the mean position or the equilibrium position. This system can stay in equilibrium if the string is vertical. Simple Pendulum A simple pendulum consists of a heavy particle suspended from a fixed support through a light, inextensible string. Thus the total mechanical energy remains constant and is independent of time. The total mechanical energy at time t is E = U + K The kinetic energy at a time t is K = ½ mv 2 Let us choose the origin to be zero of potential energy, then U(0) = 0 and U(x) = ½ kx 2 ![]() W = 0x-kxdx = -kx 2 /2 As the change in potential energy corresponding to a force is negative of the work done by the force, U(x) – U(0) = -W = ½ kx 2 The work done in a displacement from x = 0 to x is The work done by the force F during a displacement from x to x + dx is Thus the spring-block system forms a simple harmonic oscillator with angular frequency, ω = √(k/m) and time period, T = 2п/ω = 2п√(m/k).Įnergy of SHM Simple Harmonic motion is defined by the equation F = -kx. It is Hooke’s lawį = -kx where k = mω 2 For a spring, spring constant being k = mω 2 Thus, for simple harmonic motion, F = -mAω 2 sin(ωt + ф) = -mω 2 x(t) ![]() ![]() the velocity curve.Ĭombining the equations for acceleration and displacement, we getĪ(t) = -ω 2 x(t) which implies that the acceleration is proportional to the displacement in simple harmonic motion and the two related by the square of the angular frequency.įorce Law for SHM From Newton’s second law we know that F = ma and that for SHM, a = -Aω 2 sin(ωt + ф). Also the acceleration curve is shifted to the left by 90 o w.r.t. Thus the acceleration of the particle varies between the limits -Aω 2 and +Aω 2. When the magnitude of displacement is least, that of the velocity is maximum and vice versa.Īcceleration Knowing the velocity we can arrive at acceleration for simple harmonic motion by differentiating the velocity, i.e. Also as we know that sin(90 + θ) = cosθ, we can conclude that the velocity curve is shifted to the left by 90 o w.r.t. time i.e.įrom the expression it can be seen that the velocity varies between the extreme values –Aω and +Aω. Velocity The expression for velocity can be derived by differentiating the displacement w.r.t. the number of radians covered per unit time (where 1 complete oscillation corresponds to 2п radians). the number of oscillations per unit time, (2п/T) is the angular frequency i.e. We know that the motion repeats after a certain time called the time period (T) of the motion i.e x should have the same value at time t and t+T.Īlso, the velocity should have the same value at t+T, i.e.Īs T is the smallest time for repetition both the above equations are true for ωT = 2п, n = 1,2,3,……Īs (1/T) is the frequency of oscillation i.e. But on the other hand, if for the convenience of the problem, t = 0 has to be considered at a point when the particle at an extreme position, then ф would be ±п/2 depending on the direction of velocity of the particle at that position.Īngular frequency (ω) To understand angular frequency, we first try to understand frequency of simple harmonic motion. If we choose to call the instant when the particle is passing through its mean position as the time t = 0, then ф = 0 as at that time x has to be zero. To describe the motion quantitatively, a particular instant should be called zero and measurement of time should be made from this instant. It depends on the choice of the instant t = 0. Phase Angle (ф) The time varying quantity (ωt +ф) is called the phase of the motion and the quantity ф is called the phase angle or phase constant. It is the maximum displacement of the particle from the center of oscillation i.e. This gives the physical significance of the constant A. As sin(ωt +ф) can take values between -1 and +1, the displacement x can take values between –A and +A. ![]() Amplitude The quantity A is called the amplitude of the motion. ![]()
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