1. a = -ky
2.
Consider a particle moving along the circumference of a circle of radius ‘a’ and center O, with uniform angular velocity w. AB and CD are two mutually perpendicular diameters along X and Y axis. At time t = 0.
let the particle be at P0 so that ∠P0OB = Φ. After time ‘t’ second, let the particle reach P so that ∠POP0 = ω t. N is the foot of the perpendicular drawn from P on the diameter CD.
Similarly M is the foot of the perpendicular drawn from P to the diameter AB. When the particle moves along the circumference of the circle, the foot of the perpendicular executes to and for motion along the diameter CD or AB with O as the mean position.
From the right angle triangle O MP, we get
Cos (ωt + Φ) = OMOP
∴ OM = OPcos(ωt + Φ)
X= a cos (ωt + Φ) _______(1)
Similarly, we get
Sin (ωt + Φ) = ya (or)
Y = a sin (ωt + Φ) _______(2)
Equation (1) and (2) are similar to equations of S.H.M. The equation(1) and (2) shows that the projection of uniform circular motion on any diameter is S.H.M.
3. KE = PE
\(\frac{1}{2}\)mω2(a2 – x2) = \(\frac{1}{2}\) = mω2x2
Solving we get, x = \(\frac{a^2}{√2}\)
where a is the amplitude of oscillation.