Standard +0.3 This is a straightforward circular motion problem requiring students to relate tension to centripetal force (T = mv²/r), find maximum speed from maximum tension, then calculate period using T = 2πr/v. It involves standard formula application with clear given values and a single conceptual step, making it slightly easier than average for Further Maths mechanics.
5 One end of a light inextensible string of length 3.5 m is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass \(0.45 \mathrm {~kg} . P\) moves with constant speed in a circular path on the plane with the string taut.
The string will break if the tension in it exceeds 70 N .
Determine the minimum possible time in which \(P\) can describe a complete circle about \(O\).
Use of correct form for centripetal acceleration (soi); NB \(a = 155.55...\); do not allow for conical pendulum
\(70 = 0.45v_{\max}^2/3.5\) or \(0.45\times3.5\times\omega_{\max}^2\)
M1 (AO 3.1b)
Use of NII with their \(a\); forces must all be horizontal
\(70/3 = 2\pi\times3.5/T_{\min}\) or \(20/3 = 2\pi/T_{\min}\)
M1 (AO 1.1)
Use of correct formula to relate \(v\) or \(\omega\) to the period; from \(v_{\max} = 70/3\) (or awrt 23.3) or \(\omega_{\max} = 20/3\) (or awrt 6.67)
So minimum time is awrt \(0.942\) s
A1 [4] (AO 1.1)
\(3\pi/10\); SC2 for use of conical pendulum leading to correct answer (SC1 if correct to 2sf (0.94))
# Question 5:
$a = v^2/r$ or $r\omega^2$ or $v\omega$ | B1 (AO 1.2) | Use of correct form for centripetal acceleration (soi); NB $a = 155.55...$; do not allow for conical pendulum
$70 = 0.45v_{\max}^2/3.5$ or $0.45\times3.5\times\omega_{\max}^2$ | M1 (AO 3.1b) | Use of NII with their $a$; forces must all be horizontal
$70/3 = 2\pi\times3.5/T_{\min}$ or $20/3 = 2\pi/T_{\min}$ | M1 (AO 1.1) | Use of correct formula to relate $v$ or $\omega$ to the period; from $v_{\max} = 70/3$ (or awrt 23.3) or $\omega_{\max} = 20/3$ (or awrt 6.67)
So minimum time is awrt $0.942$ s | A1 [4] (AO 1.1) | $3\pi/10$; SC2 for use of conical pendulum leading to correct answer (SC1 if correct to 2sf (0.94))
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5 One end of a light inextensible string of length 3.5 m is attached to a fixed point $O$ on a smooth horizontal plane. The other end of the string is attached to a particle $P$ of mass $0.45 \mathrm {~kg} . P$ moves with constant speed in a circular path on the plane with the string taut.
The string will break if the tension in it exceeds 70 N .
Determine the minimum possible time in which $P$ can describe a complete circle about $O$.
\hfill \mbox{\textit{OCR Further Mechanics AS 2022 Q5 [4]}}