Challenging +1.2 This is a standard rotational dynamics problem requiring application of Newton's second law to both the block and disc, plus the rotational equation of motion. While it involves multiple connected components and the moment of inertia of a disc (½ma²), the setup is straightforward with clearly defined forces. The 5-mark allocation and systematic approach (equations for block, disc, and constraint relating linear/angular acceleration) make it a moderately above-average question but not requiring novel insight.
\includegraphics{figure_1}
A uniform disc with centre \(O\), mass \(m\) and radius \(a\) is free to rotate without resistance in a vertical plane about a horizontal axis through \(O\). One end of a light inextensible string is attached to the rim of the disc and wrapped around the rim. The other end of the string is attached to a block of mass \(3m\) (see diagram). The system is released from rest with the block hanging vertically. While the block is in motion, it experiences a constant vertical resisting force of magnitude \(0.9mg\). Find the tension in the string in terms of \(m\) and \(g\). [5]
3mg – 0⋅9mg – T = 3 m × a d2θ / dt2or 3m d2x / dt2
M1 A1)
Find eqn of motion for block
OR:
Answer
Marks
Guidance
Taθ = ½ (½ ma2) (dθ / dt)2
(B1
Find eqn. of energy for disc (in terms of θ or x)
(3mg – 0⋅9mg – T) aθ = ½ 3 m (a dθ / dt)2
M1 A1)
Find eqn of energy for block (in terms of θ or x)
6T = 2⋅1mg – T, T = 0⋅3 mg
M1 A1
Combine two eqns to find T
Total:
5
Question 1:
1 | EITHER:
T × a = (½ ma2) d2θ / dt2or (½ ma) d2x / dt2 | (B1 | Find eqn of motion for disc
3mg – 0⋅9mg – T = 3 m × a d2θ / dt2or 3m d2x / dt2 | M1 A1) | Find eqn of motion for block
OR:
Taθ = ½ (½ ma2) (dθ / dt)2 | (B1 | Find eqn. of energy for disc (in terms of θ or x)
(3mg – 0⋅9mg – T) aθ = ½ 3 m (a dθ / dt)2 | M1 A1) | Find eqn of energy for block (in terms of θ or x)
6T = 2⋅1mg – T, T = 0⋅3 mg | M1 A1 | Combine two eqns to find T
Total: | 5
\includegraphics{figure_1}
A uniform disc with centre $O$, mass $m$ and radius $a$ is free to rotate without resistance in a vertical plane about a horizontal axis through $O$. One end of a light inextensible string is attached to the rim of the disc and wrapped around the rim. The other end of the string is attached to a block of mass $3m$ (see diagram). The system is released from rest with the block hanging vertically. While the block is in motion, it experiences a constant vertical resisting force of magnitude $0.9mg$. Find the tension in the string in terms of $m$ and $g$. [5]
\hfill \mbox{\textit{CAIE FP2 2017 Q1 [5]}}