Challenging +1.2 This is a multi-step energy conservation problem involving elastic strings with vertical motion. Students must identify the natural length position, calculate elastic potential energy at two positions, apply conservation of energy including gravitational PE, and handle the transition from string to slack. While requiring careful bookkeeping of energy terms and understanding of when the string is taut vs slack, it follows a standard Further Mechanics template without requiring novel geometric insight or proof techniques.
1 One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to a fixed point \(O\). The other end is attached to a particle \(P\) of mass 0.4 kg . \(O\) is a vertical distance of 1 m below a horizontal ceiling. \(P\) is held at a point 1.5 m vertically below \(O\) and released from rest (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-2_470_371_593_255}
Assuming that there is no obstruction to the motion of \(P\) as it passes \(O\), find the speed of \(P\) when it first hits the ceiling.
Question 1:
1 | 24×0.92
Initial Elastic PE = =
2×0.6
24×0.42
Final Elastic PE = =
2×0.6
Increase in PE = 0.4g×2.5
“16.2” = “3.2” + ½×0.4v2 + “9.8”
v2 = 16 => speed is 4ms–1 | B1
B1
M1
M1
A1
[5] | 1.1
1.1
1.1
1.1
1.1 | λx2
Use of with attempt at
2l
finding extension (ie not just
x = 1.5)
λx2
Use of with attempt at
2l
finding extension (ie not just
x = 1)
Attempt at use of “mgh” to find
the increase of gravitational PE
from initial position to ceiling
Attempt at conservation of
energy with consideration of KE
and their PE
Not ±. Units required. | 16.2 J
3.2 J
9.8 J
8.624 J
1 One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to a fixed point $O$. The other end is attached to a particle $P$ of mass 0.4 kg . $O$ is a vertical distance of 1 m below a horizontal ceiling. $P$ is held at a point 1.5 m vertically below $O$ and released from rest (see diagram).\\
\includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-2_470_371_593_255}
Assuming that there is no obstruction to the motion of $P$ as it passes $O$, find the speed of $P$ when it first hits the ceiling.
\hfill \mbox{\textit{OCR Further Mechanics 2021 Q1 [5]}}