Horizontal elastic string on rough surface - Hooke's law and elastic energy

CAIE M2 2005 June Q4

7 marks Standard +0.8

4 A particle $P$ of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point $O$ on a rough horizontal table. $P$ is released from rest at a point on the table 3.5 m from $O$. The speed of $P$ at the instant the string becomes slack is $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find

the work done against friction during the period from the release of $P$ until the string becomes slack,
the coefficient of friction between $P$ and the table.

CAIE M2 2004 November Q5

7 marks Standard +0.8

5 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 16 N is attached to a fixed point $O$ of a horizontal table. A particle $P$ of mass 0.8 kg is attached to the other end of the string. The particle $P$ is released from rest on the table, at a point which is 0.5 m from $O$. The coefficient of friction between the particle and the table is 0.2 . By considering work and energy, find the speed of $P$ at the instant the string becomes slack.

CAIE Further Paper 3 2022 June Q2

5 marks Challenging +1.2

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 $\frac { 4 } { 3 } \mathrm { mg }$. The other end of the string is attached to a fixed point $O$ on a rough horizontal surface. The particle is at rest on the surface with the string at its natural length. The coefficient of friction between $P$ and the surface is $\frac { 1 } { 3 }$. The particle is projected along the surface in the direction $O P$ with a speed of $\frac { 1 } { 2 } \sqrt { \mathrm { ga } }$. Find the greatest extension of the string during the subsequent motion.

Edexcel M3 2014 January Q2

9 marks Standard +0.8

2. A particle $P$ of mass $m$ is attached to one end of a light elastic spring, of natural length $l$ and modulus of elasticity $2 m g$. The other end of the spring is attached to a fixed point $A$ on a rough horizontal plane. The particle is held at rest on the plane at a point $B$, where $A B = \frac { 1 } { 2 } l$, and released from rest. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 4 }$ Find the distance of $P$ from $B$ when $P$ first comes to rest.

Edexcel M3 2017 January Q5

9 marks Standard +0.8

A particle $P$ of mass $4 m$ is attached to one end of a light elastic string of natural length $l$ and modulus of elasticity 3 mg . The other end of the string is attached to a fixed point $O$ on a rough horizontal table. The particle lies at rest at the point $A$ on the table, where $O A = \frac { 4 } { 3 } l$. The coefficient of friction between $P$ and the table is $\mu$.
1. Show that $\mu \geqslant \frac { 1 } { 4 }$
The particle is now moved along the table to the point $B$, where $O B = 2 l$, and released from rest. Given that $\mu = \frac { 2 } { 5 }$
show that $P$ comes to rest before the string becomes slack.
(5)

Edexcel M3 2021 June Q6

14 marks Standard +0.8

A particle $P$ of mass 0.4 kg is attached to one end of a light elastic string, of natural length 0.8 m and modulus of elasticity 0.6 N . The other end of the string is fixed to a point $A$ on a rough horizontal table. The coefficient of friction between $P$ and the table is $\frac { 1 } { 7 }$

The particle $P$ is projected from $A$, with speed $1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, along the surface of the table.
After travelling 0.8 m from $A$, the particle passes through the point $B$ on the table.

Find the speed of $P$ at the instant it passes through $B$. The particle $P$ comes to rest at the point $C$ on the table, where $A B C$ is a straight line.
Find the total distance travelled by $P$ as it moves directly from $A$ to $C$.
Show that $P$ remains at rest at $C$.

Edexcel M3 2007 January Q3

9 marks Standard +0.8

3. A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $a$ and modulus of elasticity 3.6 mg . The other end of the string is fixed at a point $O$ on a rough horizontal table. The particle is projected along the surface of the table from $O$ with speed $\sqrt { } ( 2 a g )$. At its furthest point from $O$, the particle is at the point $A$, where $O A = \frac { 4 } { 3 } a$.

Find, in terms of $m , g$ and $a$, the elastic energy stored in the string when $P$ is at $A$.
Using the work-energy principle, or otherwise, find the coefficient of friction between $P$ and the table.

Edexcel M3 2011 June Q5

12 marks Standard +0.8

A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $l$ and modulus of elasticity $3 m g$. The other end of the string is attached to a fixed point $O$ on a rough horizontal table. The particle lies at rest at the point $A$ on the table, where $O A = \frac { 7 } { 6 } l$. The coefficient of friction between $P$ and the table is $\mu$.
1. Show that $\mu \geqslant \frac { 1 } { 2 }$.
The particle is now moved along the table to the point $B$, where $O B = \frac { 3 } { 2 } l$, and released from rest. Given that $\mu = \frac { 1 } { 2 }$, find
the speed of $P$ at the instant when the string becomes slack,
the total distance moved by $P$ before it comes to rest again.

Edexcel M3 2013 June Q4

9 marks Standard +0.8

A particle $P$ of mass 2 kg is attached to one end of a light elastic string of natural length 1.2 m . The other end of the string is attached to a fixed point $O$ on a rough horizontal plane. The coefficient of friction between $P$ and the plane is $\frac { 2 } { 5 }$. The particle is held at rest at a point $B$ on the plane, where $O B = 1.5 \mathrm {~m}$. When $P$ is at $B$, the tension in the string is 20 N . The particle is released from rest.
1. Find the speed of $P$ when $O P = 1.2 \mathrm {~m}$.
The particle comes to rest at the point $C$.
Find the distance $B C$.

AQA M2 2012 January Q8

14 marks Standard +0.3

8 An elastic string has one end attached to a point $O$ fixed on a rough horizontal surface. The other end of the string is attached to a particle of mass 2 kg . The elastic string has natural length 0.8 metres and modulus of elasticity 32 newtons. The particle is pulled so that it is at the point $A$, on the surface, 3 metres from the point $O$.

Calculate the elastic potential energy when the particle is at the point $A$.
The particle is released from rest at the point $A$ and moves in a straight line towards $O$. The particle is next at rest at the point $B$. The distance $A B$ is 5 metres. \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-6_179_1055_877_497} Find the frictional force acting on the particle as it moves along the surface.
Show that the particle does not remain at rest at the point $B$.
The particle next comes to rest at a point $C$ with the string slack. Find the distance $B C$.
Hence, or otherwise, find the total distance travelled by the particle after it is released from the point $A$.

Edexcel M3 Q7

14 marks Challenging +1.2

7. A particle of mass 2 kg is attached to one end of a light elastic string of natural length 1 m and modulus of elasticity 50 N . The other end of the string is attached to a fixed point $O$ on a rough horizontal plane and the coefficient of friction between the particle and the plane is $\frac { 10 } { 49 }$. The particle is projected from $O$ along the plane with an initial speed of $5 \mathrm {~ms} ^ { - 1 }$.

Show that the greatest distance from $O$ which the particle reaches is 1.84 m .
Find, correct to 2 significant figures, the speed at which the particle returns to $O$.

AQA M2 2009 June Q6

12 marks Standard +0.3

6 A block, of mass 5 kg , is attached to one end of a length of elastic string. The other end of the string is fixed to a vertical wall. The block is placed on a horizontal surface. The elastic string has natural length 1.2 m and modulus of elasticity 180 N . The block is pulled so that it is 2 m from the wall and is then released from rest. Whilst taut, the string remains horizontal. It may be assumed that, after the string becomes slack, it does not interfere with the movement of the block. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-5_396_960_660_534}

Calculate the elastic potential energy when the block is 2 m from the wall.
If the horizontal surface is smooth, find the speed of the block when it hits the wall.
The surface is in fact rough and the coefficient of friction between the block and the surface is $\mu$. Find $\mu$ if the block comes to rest just as it reaches the wall.

Edexcel M3 2016 June Q3

7 marks Standard +0.8

A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $l$ and modulus of elasticity $4mg$. The other end of the string is attached to a fixed point $O$ on a rough horizontal plane. The coefficient of friction between $P$ and the plane is $\frac{2}{5}$. The particle is held at a point $A$ on the plane, where $OA = \frac{5}{4}l$, and released from rest. The particle comes to rest at the point $B$.

Show that $OB < l$ [4]
Find the distance $OB$. [3]

AQA Further Paper 1 2021 June Q15

13 marks Challenging +1.8

In this question use $g = 9.8$ m s$^{-2}$ A particle $P$ of mass $m$ is attached to two light elastic strings, $AP$ and $BP$. The other ends of the strings, $A$ and $B$, are attached to fixed points which are 4 metres apart on a rough horizontal surface at the bottom of a container. The coefficient of friction between $P$ and the surface is 0.68 • When the extension of string $AP$ is $e_A$ metres, the tension in $AP$ is $24me_A$ • When the extension of string $BP$ is $e_B$ metres, the tension in $BP$ is $10me_B$ • The natural length of string $AP$ is 1 metre • The natural length of string $BP$ is 1.3 metres \includegraphics{figure_15}

Show that when $AP = 1.5$ metres, the tension in $AP$ is equal to the tension in $BP$. [1 mark]
$P$ is held at the point between $A$ and $B$ where $AP = 1.9$ metres, and then released from rest. At time $t$ seconds after $P$ is released, $AP = (1.5 + x)$ metres. \includegraphics{figure_15b} Show that when $P$ is moving towards $A$, $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 34x = 6.664$$ [3 marks]
The container is then filled with oil, and $P$ is again released from rest at the point between $A$ and $B$ where $AP = 1.9$ metres. At time $t$ seconds after $P$ is released, the oil causes a resistive force of magnitude $10mv$ newtons to act on the particle, where $v$ m s$^{-1}$ is the speed of the particle. Find $x$ in terms of $t$ when $P$ is moving towards $A$. [9 marks]