How to find period in shm?

The oscillations of a system in which the net force can be described by Hooke’s law are of special importance, because they are very common. They are also the simplest oscillatory systems. Simple Harmonic Motion (SHM) is the name given to oscillatory motion for a system where the net force can be described by Hooke’s law, and such a system is called a simple harmonic oscillator. If the net force can be described by Hooke’s law and there is no damping (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position, as shown for an object on a spring in Figure 1. The maximum displacement from equilibrium is called the amplitude X. The units for amplitude and displacement are the same, but depend on the type of oscillation. For the object on the spring, the units of amplitude and displacement are meters; whereas for sound oscillations, they have units of pressure (and other types of oscillations have yet other units). Because amplitude is the maximum displacement, it is related to the energy in the oscillation.

What is so significant about simple harmonic motion? One special thing is that the period T and frequency f of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, will oscillate with the same frequency whether plucked gently or hard. Because the period is constant, a simple harmonic oscillator can be used as a clock.

Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is. A very stiff object has a large force constant k, which causes the system to have a smaller period. For example, you can adjust a diving board’s stiffness—the stiffer it is, the faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one.

In fact, the mass m and the force constant k are the only factors that affect the period and frequency of simple harmonic motion.

If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in Figure 2. Similarly, Figure 3 shows an object bouncing on a spring as it leaves a wavelike “trace of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.

The displacement as a function of time t in any simple harmonic motion—that is, one in which the net restoring force can be described by Hooke’s law, is given by

[latex]x(t)=X\cos\frac{2\pi{t}}{T}\\[/latex],

where X is amplitude. At t = 0, the initial position is x0 = X, and the displacement oscillates back and forth with a period T. (When t = T, we get x = X again because cos 2π = 1.). Furthermore, from this expression for x, the velocity v as a function of time is given by

[latex]v(t)=-v_{\text{max}}\sin\left(\frac{2\pi{t}}{T}\right)\\[/latex], where [latex]v_{\text{max}}=\frac{2\pi{X}}{T}=X\sqrt{\frac{k}{m}}\\[/latex].

The object has zero velocity at maximum displacement—for example, v=0 when t=0, and at that time x=X. The minus sign in the first equation for v(t) gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. Finally, we can get an expression for acceleration using Newton’s second law. [Then we have x(t), v(t), t, and a(t), the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton’s second law, the acceleration is [latex]a=\frac{F}{m}=\frac{kx}{m}\\[/latex]. So, a(t) is also a cosine function:

[latex]a(t)=-\frac{kX}{m}\cos\frac{2\pi{t}}{T}\\[/latex].

Hence, a(t) is directly proportional to and in the opposite direction to a(t).

Figure 4 shows the simple harmonic motion of an object on a spring and presents graphs of x(t), v(t), and a(t) versus time.

The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion. They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another.

• Time period is the necessary time to complete a single wave.
• The unit of time period is second.
• Time period is defined by the formulae, T = 2 π ω = 1 f , where is the angular frequency of a wave and f is its frequency.
• ω = 2 π f .

Oscillations are happening all around us, from the beating of the human heart, to the vibrating atoms that make up everything. Simple harmonic motion is a very important type of periodic oscillation where the acceleration (α) is proportional to the displacement (x) from equilibrium, in the direction of the equilibrium position.

Since simple harmonic motion is a periodic oscillation, we can measure its period (the time it takes for one oscillation) and therefore determine its frequency (the number of oscillations per unit time, or the inverse of the period).

The two most common experiments that demonstrate this are:

1. Pendulum - Where a mass m attached to the end of a pendulum of length l, will oscillate with a period (T). Described by: T = 2π√(l/g), where g is the gravitational acceleration.

2. Mass on a spring - Where a mass m attached to a spring with spring constant k, will oscillate with a period (T). Described by: T = 2π√(m/k).

By timing the duration of one complete oscillation we can determine the period and hence the frequency. Note that in the case of the pendulum, the period is independent of the mass, whilst the case of the mass on a spring, the period is independent of the length of spring. The period of a simple harmonic oscillator is also independent of its amplitude.

From its definition, the acceleration, a, of an object in simple harmonic motion is proportional to its displacement, x:

Described by: T = 2π√(l/g), where g is the gravitational acceleration.