Standard +0.3 This is a standard energy conservation problem with Hooke's law on an inclined plane. Students need to equate gravitational PE lost to elastic PE gained, set up the equation ½(3mg/a)x² = mg·x·sin(θ), and solve the resulting quadratic. It's straightforward application of familiar techniques with no conceptual surprises, making it slightly easier than average for Further Maths.
A particle \(P\) of mass \(m\) is placed on a fixed smooth plane which is inclined at an angle \(\theta\) to the horizontal. A light spring, of natural length \(a\) and modulus of elasticity \(3mg\), has one end attached to \(P\) and the other end attached to a fixed point \(O\) at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length.
Find, in terms of \(a\) and \(\theta\), an expression for the greatest extension of the spring in the subsequent motion. [3]
A particle $P$ of mass $m$ is placed on a fixed smooth plane which is inclined at an angle $\theta$ to the horizontal. A light spring, of natural length $a$ and modulus of elasticity $3mg$, has one end attached to $P$ and the other end attached to a fixed point $O$ at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length.
Find, in terms of $a$ and $\theta$, an expression for the greatest extension of the spring in the subsequent motion. [3]
\hfill \mbox{\textit{CAIE Further Paper 3 2020 Q1 [3]}}