Standard +0.3 This is a straightforward equilibrium problem requiring resolution of forces in two perpendicular directions. Students need to identify the three forces (weight, tension, applied force), use the given angles (45° and 90°), and solve two simultaneous equations. While it requires careful diagram interpretation and systematic force resolution, it's a standard M1 equilibrium question with no conceptual surprises—slightly easier than average due to the convenient angles and clear setup.
A smooth ring \(R\) of mass \(m\) kg is threaded on a light inextensible string \(ARB\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(ARB = 90°\). The angle between the part \(AR\) of the string and the vertical is \(45°\). The ring is held in equilibrium in this position by a force of magnitude \(2.5\) N, acting on the ring in the direction \(BR\) (see diagram). Calculate the tension in the string and the mass of the ring. [4]
\includegraphics{figure_1}
Resolve forces perpendicular to BR and eliminate T or m
T = 1.25 N
A1
Mass of ring = 0.177 kg
A1
Allow m = √2/8
Second alternative method for Q1
2Tcos45 2.5 mg
= =
sin135 sin90 sin135
or
2.5−T T mg
= =
Answer
Marks
Guidance
sin135 sin135 sin90
M1
Attempt to apply Lami’s theorem,
M1
All three terms of Lami attempted
T = 1.25 N
A1
Mass of ring = 0.177 kg
A1
Allow m = √2/8
4
Answer
Marks
Guidance
Question
Answer
Marks
Question 1:
1 | [T cos 45 + T cos 45 = 2.5 cos 45] | M1 | For resolving horizontally
T = 1.25 N | A1
[2.5 sin 45 = mg] | M1 | For resolving vertically
Mass of ring = 0.177 kg | A1 | Allow m = √2/8
First alternative method for Q1
[2.5 = T + mg cos 45] | M1 | Resolve forces along BR
[T = mg cos 45] | M1 | Resolve forces perpendicular to BR and eliminate T or m
T = 1.25 N | A1
Mass of ring = 0.177 kg | A1 | Allow m = √2/8
Second alternative method for Q1
2Tcos45 2.5 mg
= =
sin135 sin90 sin135
or
2.5−T T mg
= =
sin135 sin135 sin90 | M1 | Attempt to apply Lami’s theorem,
M1 | All three terms of Lami attempted
T = 1.25 N | A1
Mass of ring = 0.177 kg | A1 | Allow m = √2/8
4
Question | Answer | Marks | Guidance
A smooth ring $R$ of mass $m$ kg is threaded on a light inextensible string $ARB$. The ends of the string are attached to fixed points $A$ and $B$ with $A$ vertically above $B$. The string is taut and angle $ARB = 90°$. The angle between the part $AR$ of the string and the vertical is $45°$. The ring is held in equilibrium in this position by a force of magnitude $2.5$ N, acting on the ring in the direction $BR$ (see diagram). Calculate the tension in the string and the mass of the ring. [4]
\includegraphics{figure_1}
\hfill \mbox{\textit{CAIE M1 2018 Q1 [4]}}