Vertical circle: tension at specific point

CAIE Further Paper 3 2022 June Q4

8 marks Challenging +1.8

4 One end of a light inextensible string of length $a$ is attached to a fixed point $O$. A particle of mass $m$ is attached to the other end of the string and is held with the string taut at the point $A$. At $A$ the string makes an angle $\theta$ with the upward vertical through $O$. The particle is projected perpendicular to the string in a downward direction from $A$ with a speed $u$. It moves along a circular path in the vertical plane. When the string makes an angle $\alpha$ with the downward vertical through $O$, the speed of the particle is $2 u$ and the magnitude of the tension in the string is 10 times its magnitude at $A$. It is given that $\mathrm { u } = \sqrt { \frac { 2 } { 3 } \mathrm { ga } }$.

Find, in terms of $m$ and $g$, the magnitude of the tension in the string at $A$.
Find the value of $\cos \alpha$.

CAIE Further Paper 3 2023 November Q6

11 marks Challenging +1.8

6 A particle $P$ of mass $m$ is attached to one end of a light inextensible rod of length $3 a$. An identical particle $Q$ is attached to the other end of the rod. The rod is smoothly pivoted at a point $O$ on the rod, where $\mathrm { OQ } = \mathrm { x }$. The system, of rod and particles, rotates about $O$ in a vertical plane. At an instant when the rod is vertical, with $P$ above $Q$, the particle $P$ is moving horizontally with speed $u$. When the rod has turned through an angle of $60 ^ { \circ }$ from the vertical, the speed of $P$ is $2 \sqrt { \mathrm { ag } }$, and the tensions in the two parts of the rod, $O P$ and $O Q$, have equal magnitudes.

Show that the speed of $Q$ when the rod has turned through an angle of $60 ^ { \circ }$ from the vertical is $\frac { 2 x } { 3 a - x } \sqrt { a g }$.
Find $x$ in terms of $a$.
Find $u$ in terms of $a$ and $g$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 3 2024 November Q2

5 marks Challenging +1.2

2 A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle $P$ is held at the point $A$ with the string taut. It is given that $O A$ makes an angle $\theta$ with the downward vertical through $O$, where $\tan \theta = \frac { 3 } { 4 }$. The particle $P$ is projected perpendicular to $O A$ in an upwards direction with speed $\sqrt { 5 a g }$, and it starts to move along a circular path in a vertical plane. When $P$ is at the point $B$, where angle $A O B$ is a right angle, the tension in the string is $T$. Find $T$ in terms of $m$ and $g$.

Edexcel M3 2014 January Q3

8 marks Challenging +1.2

3. A light rod $A B$ of length $2 a$ has a particle $P$ of mass $m$ attached to $B$. The rod is rotating in a vertical plane about a fixed smooth horizontal axis through $A$. Given that the greatest tension in the rod is $\frac { 9 m g } { 8 }$, find, to the nearest degree, the angle between the rod and the downward vertical when the speed of $P$ is $\sqrt { \left( \frac { a g } { 20 } \right) }$.

Edexcel M3 2010 January Q5

11 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-10_590_858_242_575} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} One end $A$ of a light inextensible string of length $3 a$ is attached to a fixed point. A particle of mass $m$ is attached to the other end $B$ of the string. The particle is held in equilibrium at a distance $2 a$ below the horizontal through $A$, with the string taut. The particle is then projected with speed $\sqrt { } ( 2 a g )$, in the direction perpendicular to $A B$, in the vertical plane containing $A$ and $B$, as shown in Figure 4. In the subsequent motion the string remains taut. When $A B$ is at an angle $\theta$ below the horizontal, the speed of the particle is $v$ and the tension in the string is $T$.

Show that $v ^ { 2 } = 2 \operatorname { ag } ( 3 \sin \theta - 1 )$.
Find the range of values of $T$.

Edexcel M3 2009 June Q4

9 marks Challenging +1.2

The finite region bounded by the $x$-axis, the curve $y = \frac { 1 } { x ^ { 2 } }$, the line $x = \frac { 1 } { 4 }$ and the line $x = 1$, is rotated through one complete revolution about the $x$-axis to form a uniform solid of revolution.
1. Show that the volume of the solid is $21 \pi$.
2. Find the coordinates of the centre of mass of the solid.
3. One end of a light inextensible string of length $l$ is attached to a fixed point $A$. The other end is attached to a particle $P$ of mass $m$, which is held at a point $B$ with the string taut and $A P$ making an angle $\arccos \frac { 1 } { 4 }$ with the downward vertical. The particle is released from rest. When $A P$ makes an angle $\theta$ with the downward vertical, the string is taut and the tension in the string is $T$.
4. Show that
$$T = 3 m g \cos \theta - \frac { m g } { 2 }$$ (6) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c9d14ac-0757-4cdd-9534-337e6b3acee0-08_678_629_815_653} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} At an instant when $A P$ makes an angle of $60 ^ { \circ }$ to the downward vertical, $P$ is moving upwards, as shown in Figure 3. At this instant the string breaks. At the highest point reached in the subsequent motion, $P$ is at a distance $d$ below the horizontal through $A$.
Find $d$ in terms of $l$.

OCR Further Mechanics 2024 June Q4

15 marks Standard +0.8

4 A particle, $P$, of mass 6 kg is attached to one end of a light inextensible rod of length 2.4 m . The other end of the rod is smoothly hinged at a fixed point $O$ and the rod is free to rotate in any direction. Initially, $P$ is at rest, vertically below $O$, when it is projected horizontally with a speed of $12 \mathrm {~ms} ^ { - 1 }$. It subsequently describes complete vertical circles with $O$ as the centre.
\includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_611_517_536_246} The angle that the rod makes with the downward vertical through $O$ at each instant is denoted by $\theta$ and $A$ is the point which $P$ passes through where $\theta = 40 ^ { \circ }$ (see diagram).

Find the tangential acceleration of $P$ at $A$, stating its direction.
Determine the radial acceleration of $P$ at $A$, stating its direction.
Find the magnitude of the force in the rod when $P$ is at $A$, stating whether the rod is in tension or compression. The motion is now stopped when $P$ is at $A$, and $P$ is then projected in such a way that it now describes horizontal circles at a constant speed with $\theta = 40 ^ { \circ }$ (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_403_524_1877_242}
Find the speed of $P$.
Explain why, wherever $P$ 's motion is initiated from and whatever its initial velocity, it is not possible for $P$ to describe horizontal circles at constant speed with $\theta = 90 ^ { \circ }$.

OCR M3 2007 January Q1

6 marks Standard +0.3

1 A particle $P$ of mass 0.6 kg is attached to a fixed point $O$ by a light inextensible string of length 0.4 m . While hanging at a distance 0.4 m vertically below $O , P$ is projected horizontally with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and moves in a complete vertical circle. Calculate the tension in the string when $P$ is vertically above $O$.

OCR M3 2011 January Q2

6 marks Standard +0.3

2 A particle of mass 0.4 kg is attached to a fixed point $O$ by a light inextensible string of length 0.5 m . The particle is projected horizontally with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from the point 0.5 m vertically below $O$. The particle moves in a complete circle. Find the tension in the string when

the string is horizontal,
the particle is vertically above $O$.

OCR MEI M3 2008 January Q2

19 marks Standard +0.3

2

A small ball of mass 0.01 kg is moving in a vertical circle of radius 0.55 m on the smooth inside surface of a fixed sphere also of radius 0.55 m . When the ball is at the highest point of the circle, the normal reaction between the surface and the ball is 0.1 N . Modelling the ball as a particle and neglecting air resistance, find
1. the speed of the ball when it is at the highest point of the circle,
2. the normal reaction between the surface and the ball when the vertical height of the ball above the lowest point of the circle is 0.15 m .
A small object Q of mass 0.8 kg moves in a circular path, with centre O and radius $r$ metres, on a smooth horizontal surface. A light elastic string, with natural length 2 m and modulus of elasticity 160 N , has one end attached to Q and the other end attached to O . The object Q has a constant angular speed of $\omega$ rad s $^ { - 1 }$.
1. Show that $\omega ^ { 2 } = \frac { 100 ( r - 2 ) } { r }$ and deduce that $\omega < 10$.
2. Find expressions, in terms of $r$ only, for the elastic energy stored in the string, and for the kinetic energy of Q . Show that the kinetic energy of Q is greater than the elastic energy stored in the string.
3. Given that the angular speed of Q is $6 \mathrm { rad } \mathrm { s } ^ { - 1 }$, find the tension in the string.

OCR MEI M3 2011 June Q2

18 marks Standard +0.3

2

A particle P of mass 0.2 kg is connected to a fixed point O by a light inextensible string of length 3.2 m , and is moving in a vertical circle with centre O and radius 3.2 m . Air resistance may be neglected. When P is at the highest point of the circle, the tension in the string is 0.6 N .
1. Find the speed of P when it is at the highest point.
2. For an instant when OP makes an angle of $60 ^ { \circ }$ with the downward vertical, find
  (A) the radial and tangential components of the acceleration of P ,
  (B) the tension in the string.
A solid cone is fixed with its axis of symmetry vertical and its vertex V uppermost. The semivertical angle of the cone is $36 ^ { \circ }$, and its surface is smooth. A particle Q of mass 0.2 kg is connected to V by a light inextensible string, and Q moves in a horizontal circle at constant speed, in contact with the surface of the cone, as shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-3_455_609_950_808} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The particle Q makes one complete revolution in 1.8 s , and the normal reaction of the cone on Q has magnitude 0.75 N .
1. Find the tension in the string.
2. Find the length of the string.

WJEC Further Unit 3 2019 June Q6

13 marks Standard +0.8

6. The diagram shows a rollercoaster at an amusement park where a car is projected from a launch point $O$ so that it performs a loop before instantaneously coming to rest at point $C$. The car then performs the same journey in reverse.
\includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-5_677_1733_552_166} The loop section is modelled by considering the track to be a vertical circle of radius 10 m and the car as a particle of mass $m$ kg moving on the inside surface of the circular loop. You may assume that the track is smooth. At point $A$, which is the lowest point of the circle, the car has velocity $u \mathrm {~ms} ^ { - 1 }$ such that $u ^ { 2 } = 60 g$. When the car is at point $B$ the radius makes an angle $\theta$ with the downward vertical.

Find, in terms of $\theta$ and $g$, an expression for $v ^ { 2 }$, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of the car at $B$.
Show that $R \mathrm {~N}$, the reaction of the track on the car at $B$, is given by $$R = m g ( 4 + 3 \cos \theta ) .$$
Explain why the expression for $R$ in part (b) shows that the car will perform a complete loop.
This model predicts that the car will stop at $C$ at a vertical height of 30 m above $A$. However, after the car has completed the loop, the track becomes rough and the car only reaches a point $D$ at a vertical height of 28 m above $A$. The resistance to motion of the car beyond the loop is of constant magnitude $\frac { m g } { 32 } \mathrm {~N}$. Calculate the length of the rough track between $A$ and $D$.

WJEC Further Unit 3 2024 June Q7

15 marks Standard +0.3

7. One end of a light rod of length $\frac { 5 } { 7 } \mathrm {~m}$ is attached to a fixed point $O$ and the other end is attached to a particle $P$, of mass $m \mathrm {~kg}$. The particle $P$ is projected from the point $A$, which is vertically below $O$, with a horizontal speed of $u \mathrm {~ms} ^ { - 1 }$ so that it moves in a vertical circle with centre $O$. When the rod $O P$ is inclined at an angle $\theta$ to the downward vertical, the speed of $P$ is $v \mathrm {~ms} ^ { - 1 }$ and the tension in the rod is $T \mathrm {~N}$.
\includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-16_629_593_646_735}

Show that $$v ^ { 2 } = u ^ { 2 } - 14 + 14 \cos \theta$$
Hence determine the least possible value of $u ^ { 2 }$ for the particle to reach the highest point of the circle.
Given that $u ^ { 2 } = 32 \cdot 2$,
1. find, in terms of $m$ and $\theta$, an expression for $T$,
2. calculate the range of values of $\theta$ such that the rod is exerting a thrust.
  State whether your answer to (c)(ii) would be different if the mass of the particle was reduced. Give a reason for your answer. Additional page, if required. Write the question number(s) in the left-hand margin. only

CAIE FP2 2017 June Q4

10 marks Challenging +1.2

Find the tension in the string in terms of $W$.
Find the modulus of elasticity of the string in terms of $W$.
Find the angle that the force acting on the rod at $A$ makes with the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{b10d2991-abff-4d2b-b470-1df844d1c7ee-10_442_442_260_849} A particle of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle is moving in complete vertical circles with the string taut. When the particle is at the point $P$, where $O P$ makes an angle $\alpha$ with the upward vertical through $O$, its speed is $u$. When the particle is at the point $Q$, where angle $Q O P = 90 ^ { \circ }$, its speed is $v$ (see diagram). It is given that $\cos \alpha = \frac { 4 } { 5 }$.
Show that $v ^ { 2 } = u ^ { 2 } + \frac { 14 } { 5 } a g$.
The tension in the string when the particle is at $Q$ is twice the tension in the string when the particle is at $P$.
Obtain another equation relating $u ^ { 2 } , v ^ { 2 } , a$ and $g$, and hence find $u$ in terms of $a$ and $g$.
Find the least tension in the string during the motion.

CAIE FP2 2018 November Q3

14 marks Standard +0.8

Show that $u ^ { 2 } = 2 a g$.
Find the maximum tension in the string as the particle moves from $A$ to $B$.
\includegraphics[max width=\textwidth, alt={}, center]{a092cd45-dc19-476d-adf7-0198fbb2116e-06_543_807_255_669} A uniform rod $A B$ of length $2 a$ and weight $W$ rests against a smooth horizontal peg at a point $C$ on the rod, where $A C = x$. The lower end $A$ of the rod rests on rough horizontal ground. The rod is in equilibrium inclined at an angle of $45 ^ { \circ }$ to the horizontal (see diagram). The coefficient of friction between the rod and the ground is $\mu$. The rod is about to slip at $A$.
Find an expression for $x$ in terms of $a$ and $\mu$.
Hence show that $\mu \geqslant \frac { 1 } { 3 }$.
Given that $x = \frac { 3 } { 2 } a$, find the value of $\mu$ and the magnitude of the resultant force on the rod at $A$.

OCR Further Mechanics 2021 June Q2

7 marks Standard +0.8

2 One end of a light inextensible string of length 0.75 m is attached to a particle $A$ of mass 2.8 kg . The other end of the string is attached to a fixed point $O . A$ is projected horizontally with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point 0.75 m vertically above $O$ (see Fig. 2). When $O A$ makes an angle $\theta$ with the upward vertical the speed of $A$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
$\xrightarrow [ A \text { a } ] { 6 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$ Fig. 2

Show that $v ^ { 2 } = 50.7 - 14.7 \cos \theta$.
Given that the string breaks when the tension in it reaches 200 N , find the angle that $O A$ turns through between the instant that $A$ is projected and the instant that the string breaks.