A pendulum of length 1.5 m swings through a small angle θ \theta\,θ radians from the vertical.
The horizontal displacement of the pendulum bob from the vertical is x x\,x metres, where x=1.5sinθx = 1.5\sin\thetax=1.5sinθ.
The vertical rise of the bob above its lowest point is h h\,h metres, where h=1.5(1−cosθ)h = 1.5(1 - \cos\theta)h=1.5(1−cosθ).
Use small angle approximations to show that, for small θ\thetaθ,
h≈x23\displaystyle h \approx \frac{x^2}{3}h≈3x2
Use this result to estimate the vertical rise of the bob when its horizontal displacement is 0.12 m.
State one reason why this approximation would be unreliable for a horizontal displacement of 1.2 m.