Question

Prove that using X(t)=As in( at+¢) will produce the same results for the period for the oscillations of a mass and a spring. Why do you think the cosine function was chosen?

Answer 1

To prove that X(t)=Asin(at+¢) will produce the same results for the period of oscillations of a mass and a spring as cosine function, we can use the fact that sine and cosine functions are related through a phase shift of π/2, i.e., sin(x+π/2)=cos(x).

The equation of motion for a mass m attached to a spring with spring constant k can be written as:

m(d^2x/dt^2) + kx = 0 (1)

where x represents the displacement from equilibrium position.

Assuming a solution of the form x = A*cos(ωt+φ), where A is the amplitude, ω is the angular frequency, and φ is the phase angle, and substituting it into the equation of motion (1), we get:

-mω^2A*cos(ωt+φ) + kA*cos(ωt+φ) = 0

Simplifying and rearranging, we get:

(ω^2 - k/m)*A*cos(ωt+φ) = 0

Since A ≠ 0, the only way for this equation to hold is if:

ω^2 = k/m

or

ω = √(k/m)

This is the angular frequency of the oscillations, which is related to the period T by:

T = 2π/ω = 2π√(m/k)

Now, let's assume a solution of the form x = Asin(ωt+φ), and repeat the above steps. We get:

(ω^2 - k/m)*A*sin(ωt+φ) = 0

Again, since A ≠ 0, the only way for this equation to hold is if:

ω^2 = k/m

or

ω = √(k/m)

So, the angular frequency is the same for both cosine and sine functions.

To find the period T, we can use the fact that sin(x+π/2)=cos(x), and rewrite the solution as:

x = Asin(ωt+φ) = Acos(ωt+φ-π/2)

So, the period can be found using the same formula as before:

T = 2π/ω = 2π√(m/k)

This shows that using X(t)=Asin(at+¢) will produce the same results for the period of oscillations of a mass and a spring as cosine function.

The cosine function was probably chosen because it has a starting value of 1 at t=0, which corresponds to the equilibrium position of the mass. The sine function, on the other hand, has a starting value of 0 at t=0, which doesn't correspond to the equilibrium position. However, both functions can be used interchangeably, as shown above.