To understand how the two standard ways to write the general solution to a harmonic oscillator are related.

there are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:

x(t)=acos(ωt+ϕ) and

x(t)=ccos(ωt)+ssin(ωt).

either of these equations is a general solution of a second-order differential equation (f⃗ =ma⃗ ); hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand. (some texts refer to these arbitrary constants as boundary values.)

a)

find analytic expressions for the arbitrary constants c and s in equation 2 (found in part b) in terms of the constants a and ϕ in equation 1 (found in part a), which are now considered as given parameters.

give your answers for the coefficients of cos(ωt) and sin(ωt), separated by a comma. express your answers in terms of a and ϕ.

b)

find analytic expressions for the arbitrary constants a and ϕ in equation 1 (found in part a) in terms of the constants c and s in equation 2 (found in part b), which are now considered as given parameters.

express the amplitude a and phase ϕ (separated by a comma) in terms of c and s.

a)   C = A cos φ , S = A sin φ,  b)   φ = tan⁻¹ (S / C) ,  A² = C² + S²

Explanation:

a) The two forms given are equivalent, let's start by developing the double angle

            Cos (a + b) = cos a cos b - sin a sin b

Call us

           a  = wt     and      b = φ

           x = A (cos wt cosφ- sin wt sin φ

the second equation is

           x = C cos wt + S  sin wt

Let's match

           C cos wt + S sin wt = A cos φ coswt - a sin φ sin wt

The coefficients of the cosine and breasts must be equal,

         C = A cos φ

          S = A sin φ

b) Divide the last two expressions

          tan φ = S / C

          φ = tan⁻¹ (S / C)

Let's square the two equations

         C² = A² cos² φ

         S² = A² sin² φ

Let's add

        C² + S² = A² (cos² φ + sin² φ)

The part in brackets vouchers1

        A² = C² + S²

force is based upon mass and acceleration.the statement is not always true.

it can be explained from newton's second law of motion.

as per newton's second law of motion the rate of change of momentum is directly proportional to the applied force and takes place along the direction of force.

mathematically it can be written as -

                                              ⇒      

here   k is the proportionality constant whose value is 1.

                                                            [a is the acceleration=dv/dt]

hence we see that force not only depends on mass and acceleration but also on velocity and rate of change of mass. force f will depend on mass and acceleration only when mass is constant.

hence the statement that force is based upon mass and acceleration is false.