Standard +0.3 This is a standard energy conservation problem with elastic strings requiring students to identify two positions (release and slack point), apply conservation of energy including elastic PE, gravitational PE, and kinetic energy, and solve a quadratic equation. While it involves multiple energy forms, the setup is routine for M2 students and follows a well-practiced method with clear physical states.
2 A particle of mass 0.2 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 4 N . The other end of the string is attached to a fixed point \(O\). The particle is held at a point which is \(( 0.6 + x ) \mathrm { m }\) vertically below \(O\). The particle is released from rest. In the subsequent motion the speed of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the string becomes slack. By considering energy, find the value of \(x\).
Gain in KE \(= \frac{1}{2}(0.2)(3^2)\) \((= 0.9\) J\()\)
B1
Gain in GPE \(= 0.2gx\) \((= 2x\) J\()\)
B1
Loss in EPE \(= \frac{1}{2}(4x^2)/0.6\) \((= 10x^2/3\) J\()\)
B1
\([10x^2/3 = 9/10 + 2x]\)
M1
For using gain in KE + gain in GPE = loss in EPE
\(x = 0.9\)
A1
[5]
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Gain in KE $= \frac{1}{2}(0.2)(3^2)$ $(= 0.9$ J$)$ | B1 | |
| Gain in GPE $= 0.2gx$ $(= 2x$ J$)$ | B1 | |
| Loss in EPE $= \frac{1}{2}(4x^2)/0.6$ $(= 10x^2/3$ J$)$ | B1 | |
| $[10x^2/3 = 9/10 + 2x]$ | M1 | For using gain in KE + gain in GPE = loss in EPE |
| $x = 0.9$ | A1 | **[5]** |
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2 A particle of mass 0.2 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 4 N . The other end of the string is attached to a fixed point $O$. The particle is held at a point which is $( 0.6 + x ) \mathrm { m }$ vertically below $O$. The particle is released from rest. In the subsequent motion the speed of the particle is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when the string becomes slack. By considering energy, find the value of $x$.
\hfill \mbox{\textit{CAIE M2 2009 Q2 [5]}}