Equations of SHMĬonsider a block attached to a spring on a frictionless table ( Figure 15.4). Note that the force constant is sometimes referred to as the spring constant. To derive an equation for the period and the frequency, we must first define and analyze the equations of motion. In fact, the mass m and the force constant k are the only factors that affect the period and frequency of SHM. For example, a heavy person on a diving board bounces up and down more slowly than a light one. The more massive the system is, the longer the period.
Period also depends on the mass of the oscillating system. For example, you can adjust a diving board’s stiffness-the stiffer it is, the faster it vibrates, and the shorter its period. A very stiff object has a large force constant ( k), which causes the system to have a smaller period. The period is related to how stiff the system is. Two important factors do affect the period of a simple harmonic oscillator. The string of a guitar, for example, oscillates with the same frequency whether plucked gently or hard. What is so significant about SHM? For one thing, the period T and frequency f of a simple harmonic oscillator are independent of amplitude. The greater the mass of the object is, the greater the period T.
The stiffer the spring is, the smaller the period T. The object’s maximum speed occurs as it passes through equilibrium. The mass continues in SHM that has an amplitude A and a period T. (e) The mass then continues to move in the positive direction until it stops at x = A x = A. (d) The mass now begins to accelerate in the positive x-direction, reaching a positive maximum velocity at x = 0 x = 0. (c) The mass continues to move in the negative x-direction, slowing until it comes to a stop at x = − A x = − A. (b) The mass accelerates as it moves in the negative x-direction, reaching a maximum negative velocity at x = 0 x = 0. (a) The mass is displaced to a position x = A x = A and released from rest. The position of the mass, when the spring is neither stretched nor compressed, is marked as x = 0 x = 0 and is the equilibrium position. The other end of the spring is attached to the wall.
In the above set of figures, a mass is attached to a spring and placed on a frictionless table. For the object on the spring, the units of amplitude and displacement are meters.įigure 15.3 An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. The units for amplitude and displacement are the same but depend on the type of oscillation. The maximum displacement from equilibrium is called the amplitude ( A). If the net force can be described by Hooke’s law and there is no damping (slowing down due to friction or other nonconservative forces), then a simple harmonic oscillator oscillates with equal displacement on either side of the equilibrium position, as shown for an object on a spring in Figure 15.3. This force obeys Hooke’s law F s = − k x, F s = − k x, as discussed in a previous chapter. The object oscillates around the equilibrium position, and the net force on the object is equal to the force provided by the spring. In simple harmonic motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and acts in the opposite direction of the displacement.Ī good example of SHM is an object with mass m attached to a spring on a frictionless surface, as shown in Figure 15.3.
0 kommentar(er)
