`color{blue} ✍️`In section 4.2, we defined displacement of a particle as the change in its position vector. In this chapter, we use the term displacement in a more general sense. It refers to change with time of any physical property under consideration.
`color{blue} ✍️`For example, in case of rectilinear motion of a steel ball on a surface, the distance from the starting point as a function of time is its position displacement. The choice of origin is a matter of convenience.
`color{blue} ✍️`Consider a block attached to a spring, the other end of which is fixed to a rigid wall [see Fig.14.2(a)]. Generally it is convenient to measure displacement of the body from its equilibrium position.
`color{blue} ✍️`For an oscillating simple pendulum, the angle from the vertical as a function of time may be regarded as a displacement variable [see Fig.14.2(b)].
`color{blue} ✍️`The term displacement is not always to be referred in the context of position only. There can be many other kinds of displacement variables.
`color{blue} ✍️`The voltage across a capacitor, changing with time in an a.c. circuit, is also a displacement variable. In the same way, pressure variations in time in the propagation of sound wave, the changing electric and magnetic fields in a light wave are examples of displacement in different contexts.
`color{blue} ✍️`The displacement variable may take both positive and negative values. In experiments on oscillations, the displacement is measured for different times.
`color{blue} ✍️`The displacement can be represented by a mathematical function of time. In case of periodic motion, this function is periodic in time. One of the simplest periodic functions is given by
`color{blue} {f(t) = A cos omega t}`
.................... (14.3a)
`color{blue} ✍️`If the argument of this function, `ωt`, is increased by an integral multiple of `2π` radians, the value of the function remains the same. The function `f (t )` is then periodic and its period, `T`, is given by
`color{purple} {T = (2pi)/omega}`
`color{blue} ✍️`Thus, the function `f (t)` is periodic with period `T`,
`color{purple} { f (t) = f (t+T ) }`
`color{blue} ✍️`The same result is obviously correct if we consider a sine function, `f (t ) = A sin ωt`. Further,a linear combination of sine and cosine functions like,
`color{blue} {f (t) = A sin ωt + B cos ωt}`
......................(14.3c)
`color{blue} ✍️`is also a periodic function with the same period `T`. Taking,
`color{purple} {A = D cos φ and B = D sin φ }`
`color{blue} ✍️`Eq. (14.3c) can be written as,
`color{blue} {f (t) = D sin (ωt + φ )} `
, .......................(14.3d)
`color{blue} ✍️`Here `D` and `φ` are constant given by
`color{purple} { D = sqrt(A^2 +B^2 ) "and" phi = tan^-1 (B/A)}`
`color{blue} ✍️`The great importance of periodic sine and cosine functions is due to a remarkable result proved by the French mathematician, Jean Baptiste Joseph Fourier (1768-1830): Any periodic function can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients.
`color{blue} ✍️`In section 4.2, we defined displacement of a particle as the change in its position vector. In this chapter, we use the term displacement in a more general sense. It refers to change with time of any physical property under consideration.
`color{blue} ✍️`For example, in case of rectilinear motion of a steel ball on a surface, the distance from the starting point as a function of time is its position displacement. The choice of origin is a matter of convenience.
`color{blue} ✍️`Consider a block attached to a spring, the other end of which is fixed to a rigid wall [see Fig.14.2(a)]. Generally it is convenient to measure displacement of the body from its equilibrium position.
`color{blue} ✍️`For an oscillating simple pendulum, the angle from the vertical as a function of time may be regarded as a displacement variable [see Fig.14.2(b)].
`color{blue} ✍️`The term displacement is not always to be referred in the context of position only. There can be many other kinds of displacement variables.
`color{blue} ✍️`The voltage across a capacitor, changing with time in an a.c. circuit, is also a displacement variable. In the same way, pressure variations in time in the propagation of sound wave, the changing electric and magnetic fields in a light wave are examples of displacement in different contexts.
`color{blue} ✍️`The displacement variable may take both positive and negative values. In experiments on oscillations, the displacement is measured for different times.
`color{blue} ✍️`The displacement can be represented by a mathematical function of time. In case of periodic motion, this function is periodic in time. One of the simplest periodic functions is given by
`color{blue} {f(t) = A cos omega t}`
.................... (14.3a)
`color{blue} ✍️`If the argument of this function, `ωt`, is increased by an integral multiple of `2π` radians, the value of the function remains the same. The function `f (t )` is then periodic and its period, `T`, is given by
`color{purple} {T = (2pi)/omega}`
`color{blue} ✍️`Thus, the function `f (t)` is periodic with period `T`,
`color{purple} { f (t) = f (t+T ) }`
`color{blue} ✍️`The same result is obviously correct if we consider a sine function, `f (t ) = A sin ωt`. Further,a linear combination of sine and cosine functions like,
`color{blue} {f (t) = A sin ωt + B cos ωt}`
......................(14.3c)
`color{blue} ✍️`is also a periodic function with the same period `T`. Taking,
`color{purple} {A = D cos φ and B = D sin φ }`
`color{blue} ✍️`Eq. (14.3c) can be written as,
`color{blue} {f (t) = D sin (ωt + φ )} `
, .......................(14.3d)
`color{blue} ✍️`Here `D` and `φ` are constant given by
`color{purple} { D = sqrt(A^2 +B^2 ) "and" phi = tan^-1 (B/A)}`
`color{blue} ✍️`The great importance of periodic sine and cosine functions is due to a remarkable result proved by the French mathematician, Jean Baptiste Joseph Fourier (1768-1830): Any periodic function can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients.