# Some STEP probability and mechanics questions

Some STEP probability and mechanics questions from Michael Gibson’s “mini-mock” papers. These questions can be done with relatively short working and without needing to use any great knowledge of formulas. Continue reading

# Answer to STEP III 2007 Q.9

III/2007/9 – Two small beads, A and B, each of mass m, are threaded on a smooth horizontal circular hoop of radius a and centre O. The angle θ is the acute angle determined by 2 θ = AOB.

The beads are connected by a light straight spring. The energy stored in the spring is
$mk^2a^2(\theta - \alpha)^2$
where k and α are constants satisfying k > 0 and $\frac{\pi}{4} < \alpha < \frac{\pi}{2}$
The spring is held in compression with θ = β and then released. Find the period of oscillations in the two cases that arise according to the value of θ and state the value of β for which oscillations do not occur. Continue reading

# Answer to STEP I 1999 Q.2

I/1999/2 – A point moves in the x,y plane so that the sum of the squares of its distances from the three fixed points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is always $a^{2}$.

Find the equation of the locus of the point and interpret it geometrically.

Explain why $a^{2}$ cannot be less than the sum of the squares of the distances of the three points from their centroid. Continue reading