The motion of a mass on a spring, with spring constant K is as shown in figure.
The equation of motion is given by
x(t) = A sinωt + B cosωt with ω = √(K/m)
Suppose that at time t = 0, the position of mass is x(0) and velocity v(0), then its displacement can also be represented as x(t) = C cos(ωt - φ), where C and φ are :

Solution

Physics sir · 3 steps

Expand the cosine, match it term by term with A sin ωt + B cos ωt, and let x(0), v(0) fix A and B.

1. Find A and B. Put t = 0 in x(t) = A sin ω t + B cos ω t to get x(0) = B. Differentiate to get v(t) = Aω cos ω t - Bω sin ω t, so v(0) = Aω, which gives A = v(0)/ω.
2. Expand the cosine. C cos(ω t - φ) = C sin φ sin ω t + C cos φ cos ω t. Matching coefficients gives A = C sin φ and B = C cos φ.

   C = √(A² + B²) = √(v(0)²/ω² + x(0)²), tan φ = A/B = v(0)/(x(0) ω)

3. So. Squaring and adding gives C, and dividing A by B gives φ = tan⁻¹(v(0)/x(0)ω). This is option (c).

The idea to keep. Any A sin ω t + B cos ω t is a single cosine of amplitude √(A²+B²). With C cos(ω t - φ), tan φ is the sine coefficient over the cosine coefficient.

Answer from the NTA final answer key. Solution by PrepWiser.