Two strings, two fixed points - Circular Motion 1

Edexcel FM2 AS 2019 June Q3

9 marks Challenging +1.2

A light inextensible string has length $8 a$. One end of the string is attached to a fixed point $A$ and the other end of the string is attached to a fixed point $B$, with $A$ vertically above $B$ and $A B = 4 a$. A small ball of mass $m$ is attached to a point $P$ on the string, where $A P = 5 a$.

The ball moves in a horizontal circle with constant speed $v$, with both $A P$ and $B P$ taut.
The string will break if the tension in it exceeds $\frac { 3 m g } { 2 }$ By modelling the ball as a particle and assuming the string does not break,

show that $\frac { 9 a g } { 4 } < v ^ { 2 } \leqslant \frac { 27 a g } { 4 }$
find the least possible time needed for the ball to make one complete revolution.

Edexcel FM2 AS Specimen Q2

16 marks Challenging +1.2

A light inextensible string has length 7a. One end of the string is attached to a fixed point $A$ and the other end of the string is attached to a fixed point $B$, with $A$ vertically above $B$ and $A B = 5 a$. A particle of mass $m$ is attached to a point $P$ on the string where $A P = 4 a$. The particle moves in a horizontal circle with constant angular speed $\omega$, with both $A P$ and $B P$ taut.
1. Show that
  1. the tension in $A P$ is $\frac { 4 m } { 25 } \left( 9 a \omega ^ { 2 } + 5 g \right)$
  2. the tension in $B P$ is $\frac { 3 m } { 25 } \left( 16 a \omega ^ { 2 } - 5 g \right)$.
The string will break if the tension in it reaches a magnitude of $4 m g$.
The time for the particle to make one revolution is $S$.
Show that $$3 \pi \sqrt { \frac { a } { 5 g } } < S < 8 \pi \sqrt { \frac { a } { 5 g } }$$
State how in your calculations you have used the assumption that the string is light.

CAIE M2 2010 June Q5

8 marks Standard +0.8

It is given that when the ball moves with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the tension in the string $Q B$ is three times the tension in the string $P B$. Calculate the radius of the circle. The ball now moves along this circular path with the minimum possible speed.
State the tension in the string $P B$ in this case, and find the speed of the ball.

OCR M2 2008 January Q6

11 marks Standard +0.3

Show that the tension in the string is 4.16 N , correct to 3 significant figures.
Calculate $\omega$.
(ii) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_510_417_1238_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The lower part of the string is now attached to a point $R$, vertically below $P$. $P B$ makes an angle $30 ^ { \circ }$ with the vertical and $R B$ makes an angle $60 ^ { \circ }$ with the vertical. The bead $B$ now moves in a horizontal circle of radius 1.5 m with constant speed $v _ { \mathrm { m } } \mathrm { m } ^ { - 1 }$ (see Fig. 2).
1. Calculate the tension in the string.
2. Calculate $v$.

AQA M2 2009 June Q4

8 marks Standard +0.3

4 Two light inextensible strings each have one end attached to a particle, $P$, of mass 6 kg . The other ends of the strings are attached to the fixed points $B$ and $C$. The point $C$ is vertically above the point $B$. The particle moves, at constant speed, in a horizontal circle, with centre 0.6 m below point $B$, with the strings inclined at $40 ^ { \circ }$ and $60 ^ { \circ }$ to the vertical, as shown in the diagram. Both strings are taut. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-4_761_542_539_751}

As the particle moves in the horizontal circle, the tensions in the two strings are equal. Show that the tension in the strings is 46.4 N , correct to three significant figures.
Find the speed of the particle.

CAIE M2 2018 June Q6

9 marks Standard +0.8

\includegraphics{figure_6} A particle $P$ of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m. The other end of the string is attached to a fixed point $A$. The particle $P$ is also attached to one end of a second light inextensible string of length 0.6 m, the other end of which is attached to a fixed point $B$ vertically below $A$. The particle moves in a horizontal circle of radius 0.3 m, which has its centre at the mid-point of $AB$, with both strings straight (see diagram).

Calculate the least possible angular speed of $P$. [4]
Find the greatest possible speed of $P$. [5]

The string $AP$ will break if its tension exceeds 8 N. The string $BP$ will break if its tension exceeds 5 N.

CAIE M2 2017 March Q5

7 marks Standard +0.3

\includegraphics{figure_5} Two particles $P$ and $Q$ have masses $0.4 \text{ kg}$ and $m \text{ kg}$ respectively. $P$ is attached to a fixed point $A$ by a light inextensible string of length $0.5 \text{ m}$ which is inclined at an angle of $60°$ to the vertical. $P$ and $Q$ are joined to each other by a light inextensible vertical string. $Q$ is attached to a fixed point $B$, which is vertically below $A$, by a light inextensible string. The string $BQ$ is taut and horizontal. The particles rotate in horizontal circles about an axis through $A$ and $B$ with constant angular speed $\omega \text{ rad s}^{-1}$ (see diagram). The tension in the string joining $P$ and $Q$ is $1.5 \text{ N}$.

Find the tension in the string $AP$ and the value of $\omega$. [4]
Find $m$ and the tension in the string $BQ$. [3]

Edexcel M3 2011 January Q5

10 marks Standard +0.3

\includegraphics{figure_3} A small ball $P$ of mass $m$ is attached to the ends of two light inextensible strings of length $l$. The other ends of the strings are attached to fixed points $A$ and $B$, where $A$ is vertically above $B$. Both strings are taut and $AP$ is perpendicular to $BP$ as shown in Figure 3. The system rotates about the line $AB$ with constant angular speed $\omega$. The ball moves in a horizontal circle.

Find, in terms of $m$, $g$, $l$ and $\omega$, the tension in $AP$ and the tension in $BP$. [8]
Show that $\omega^2 \geq \frac{g\sqrt{2}}{l}$. [2]

Edexcel M3 2001 June Q6

14 marks Standard +0.3

\includegraphics{figure_4} A particle $P$ of mass $m$ is attached to two light inextensible strings. The other ends of the string are attached to fixed points $A$ and $B$. The point $A$ is a distance $h$ vertically above $B$. The system rotates about the line $AB$ with constant angular speed $\omega$. Both strings are taut and inclined at $60°$ to $AB$, as shown in Fig. 4. The particle moves in a circle of radius $r$.

Show that $r = \frac{\sqrt{3}}{2}h$. [2]
Find, in terms of $m$, $g$, $h$ and $\omega$, the tension in $AP$ and the tension in $BP$. [8]

The time taken for $P$ to complete one circle is $T$.

Show that $T < \pi\sqrt{\left(\frac{2h}{g}\right)}$. [4]

Edexcel M3 2003 June Q4

11 marks Standard +0.3

\includegraphics{figure_1} A particle $P$ of mass $m$ is attached to the ends of two light inextensible strings $AP$ and $BP$ each of length $l$. The ends $A$ and $B$ are attached to fixed points, with $A$ vertically above $B$ and $AB = \frac{3}{4}l$, as shown in Fig. 1. The particle $P$ moves in a horizontal circle with constant angular speed $\omega$. The centre of the circle is the mid-point of $AB$ and both strings remain taut.

Show that the tension $AP$ is $\frac{1}{6}m(3l\omega^2 + 4g)$. [7]
Find, in terms of $m$, $l$, $\omega$ and $g$, an expression for the tension in $BP$. [2]
Deduce that $\omega^2 \geq \frac{4g}{3l}$. [2]

Edexcel M3 2012 June Q3

10 marks Standard +0.3

\includegraphics{figure_1} A particle $Q$ of mass 5 kg is attached by two light inextensible strings to two fixed points $A$ and $B$ on a vertical pole. Each string has length 0.6 m and $A$ is 0.4 m vertically above $B$, as shown in Figure 1. Both strings are taut and $Q$ is moving in a horizontal circle with constant angular speed 10 rad s$^{-1}$. Find the tension in

$AQ$,
$BQ$. [10]