Elastic string in circular motion

CAIE M2 2017 June Q6

9 marks Standard +0.8

6 A particle $P$ of mass 0.15 kg is attached to one end of a light elastic string of natural length 0.4 m and modulus of elasticity 12 N . The other end of the string is attached to a fixed point $A$. The particle $P$ moves in a horizontal circle which has its centre vertically below $A$, with the string inclined at $\theta ^ { \circ }$ to the vertical and $A P = 0.5 \mathrm {~m}$.

Find the angular speed of $P$ and the value of $\theta$.
Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of $P$.
$7 \quad$ A particle $P$ of mass 0.5 kg is at rest at a point $O$ on a rough horizontal surface. At time $t = 0$, where $t$ is in seconds, a horizontal force acting in a fixed direction is applied to $P$. At time $t \mathrm {~s}$ the magnitude of the force is $0.6 t ^ { 2 } \mathrm {~N}$ and the velocity of $P$ away from $O$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It is given that $P$ remains at rest at $O$ until $t = 0.5$.
Calculate the coefficient of friction between $P$ and the surface, and show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 1.2 t ^ { 2 } - 0.3 \quad \text { for } t > 0.5$$
Express $v$ in terms of $t$ for $t > 0.5$.
Find the displacement of $P$ from $O$ when $t = 1.2$.

CAIE Further Paper 3 2024 June Q2

6 marks Challenging +1.8

2 A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $2 m g$. A particle $Q$ of mass $k m$ is attached to the other end of the string. Particle $P$ lies on a smooth horizontal table. The string has part of its length in contact with the table and then passes through a small smooth hole $H$ in the table. Particle $P$ moves in a horizontal circle on the surface of the table with constant speed $\sqrt { \frac { 1 } { 2 } g a }$. Particle $Q$ hangs in equilibrium vertically below the hole with $H Q = \frac { 1 } { 4 } a$.

Find, in terms of $a$, the extension in the string.
Find the value of $k$.

CAIE M2 2014 November Q7

12 marks Challenging +1.8

7
\includegraphics[max width=\textwidth, alt={}, center]{84a2b2eb-a750-4047-864b-4a165fc66b2a-4_558_857_260_644} One end of a light elastic string with modulus of elasticity 15 N is attached to a fixed point $A$ which is 2 m vertically above a fixed small smooth ring $R$. The string has natural length 2 m and it passes through $R$. The other end of the string is attached to a particle $P$ of mass $m \mathrm {~kg}$ which moves with constant angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$ in a horizontal circle which has its centre 0.4 m vertically below the ring. $P R$ makes an acute angle $\theta$ with the vertical (see diagram).

Show that the tension in the string is $\frac { 3 } { \cos \theta } \mathrm {~N}$ and hence find the value of $m$.
Show that the value of $\omega$ does not depend on $\theta$. It is given that for one value of $\theta$ the elastic potential energy stored in the string is twice the kinetic energy of $P$.
Find this value of $\theta$.

Edexcel M3 Q1

7 marks Standard +0.3

A motorcyclist rides in a cylindrical well of radius 5 m . He maintains a horizontal circular path at a constant speed of $10 \mathrm {~ms} ^ { - 1 }$. The coefficient of friction between the wall and the wheels of the cycle is $\mu$.
Modelling the cyclist and his machine as a particle in contact
\includegraphics[max width=\textwidth, alt={}, center]{d1acee30-26a7-44d0-b4f5-ab721451b8b8-1_359_263_370_1595}
with the wall, show that he will not slip downwards provided that $\mu \geq 0.49$.
A particle $P$ moves with simple harmonic motion in a straight line. The centre of oscillation is $O$. When $P$ is at a distance 1 m from $O$, its speed is $8 \mathrm {~ms} ^ { - 1 }$. When it is at a distance 2 m from $O$, its speed is $4 \mathrm {~ms} ^ { - 1 }$.
1. Find the amplitude of the motion.
2. Show that the period of motion is $\frac { \pi } { 2 } \mathrm {~s}$.
3. A particle of mass $m \mathrm {~kg}$ is attached to the end $B$ of a light elastic string $A B$. The string has natural length $l \mathrm {~m}$ and modulus of elasticity $\lambda . \mathrm { N }$.
The end $A$ is attached to a fixed point on a smooth plane
\includegraphics[max width=\textwidth, alt={}, center]{d1acee30-26a7-44d0-b4f5-ab721451b8b8-1_289_543_1329_1425}
inclined at an angle $\alpha$ to the horizontal, as shown, and the particle rests in equilibrium with the length $A B = \frac { 5 l } { 4 } \mathrm {~m}$.