Inverse power force - non-gravitational context

CAIE M2 2004 June Q3

7 marks Standard +0.3

3 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_145_792_1656_680} A particle $P$ of mass 0.6 kg moves in a straight line on a smooth horizontal surface. A force of magnitude $\frac { 3 } { x ^ { 3 } }$ newtons acts on the particle in the direction from $P$ to $O$, where $O$ is a fixed point of the surface and $x \mathrm {~m}$ is the distance $O P$ (see diagram). The particle $P$ is released from rest at the point where $x = 10$. Find the speed of $P$ when $x = 2.5$.

CAIE M2 2011 June Q6

9 marks Challenging +1.2

6 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-3_151_949_1206_598} $O$ and $A$ are fixed points on a horizontal surface, with $O A = 0.5 \mathrm {~m}$. A particle $P$ of mass 0.2 kg is projected horizontally with speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from $A$ in the direction $O A$ and moves in a straight line (see diagram). At time $t \mathrm {~s}$ after projection, the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its displacement from $O$ is $x \mathrm {~m}$. The coefficient of friction between the surface and $P$ is 0.5 , and a force of magnitude $\frac { 0.4 } { x ^ { 2 } } \mathrm {~N}$ acts on $P$ in the direction $P O$.

Show that, while the particle is in motion, $v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \left( 5 + \frac { 2 } { x ^ { 2 } } \right)$.
Calculate the distance travelled by $P$ before it comes to rest, and show that $P$ does not subsequently move.

CAIE M2 2019 November Q3

6 marks Challenging +1.2

3 A smooth horizontal surface has two fixed points $O$ and $A$ which are 0.8 m apart. A particle $P$ of mass 0.25 kg is projected with velocity $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ horizontally from $A$ in the direction away from $O$. The velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when the displacement of $P$ from $O$ is $x \mathrm {~m}$. A force of magnitude $k v ^ { 2 } x ^ { - 2 } \mathrm {~N}$ opposes the motion of $P$.

Show that $v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 4 k v ^ { 2 } x ^ { - 2 }$.
Express $v$ in terms of $k$ and $x$.

Edexcel M3 2018 June Q5

13 marks Standard +0.8

A particle $P$ of mass 0.8 kg moves along the $x$-axis in the positive $x$ direction under the action of a resultant force. This force acts in the direction of $x$ increasing. At time $t$ seconds, $t \geqslant 0 , P$ is $x$ metres from the origin $O , P$ is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the force has magnitude $\frac { 4 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}$.

When $t = 0 , P$ is at rest at $O$.

Show that $v ^ { 2 } = 5 \left( \frac { ( x + 1 ) ^ { 2 } - 1 } { ( x + 1 ) ^ { 2 } } \right)$ When $t = 2 , P$ is at the point $A$. When $t = 4 , P$ is at the point $B$.
Using algebraic integration, find the distance $A B$.

Edexcel M3 2020 June Q5

12 marks Standard +0.8

5. A particle $P$ of mass 0.5 kg moves along the positive $x$-axis in the positive $x$ direction. At time $t$ seconds, $t \geqslant 1 , P$ is $x$ metres from the origin $O$ and is moving with speed $v \mathrm {~ms} ^ { - 1 }$. The resultant force acting on $P$ has magnitude $\frac { 2 } { x ^ { 3 } } \mathrm {~N}$ and is directed towards $O$. When $t = 1 , x = 1$ and $v = 3$ Show that

$v ^ { 2 } = \frac { 4 } { x ^ { 2 } } + 5$
$t = \frac { a + \sqrt { b x ^ { 2 } + c } } { d }$, where $a , b , c$ and $d$ are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-13_2255_50_314_34}

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel M3 2008 January Q2

8 marks Standard +0.8

2. A particle $P$ of mass 0.1 kg moves in a straight line on a smooth horizontal table. When $P$ is a distance $x$ metres from a fixed point $O$ on the line, it experiences a force of magnitude $\frac { 16 } { 5 x ^ { 2 } } \mathrm {~N}$ away from $O$ in the direction $O P$. Initially $P$ is at a point 2 m from $O$ and is moving towards $O$ with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the distance of $P$ from $O$ when $P$ first comes to rest.

Edexcel M3 2008 June Q6

14 marks Standard +0.8

A particle $P$ of mass 0.5 kg moves along the positive $x$-axis. It moves away from the origin $O$ under the action of a single force directed away from $O$. When $O P = x$ metres, the magnitude of the force is $\frac { 3 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}$ and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Initially $P$ is at rest at $O$.
1. Show that $v ^ { 2 } = 6 \left( 1 - \frac { 1 } { ( x + 1 ) ^ { 2 } } \right)$.
2. Show that the speed of $P$ never reaches $\sqrt { } 6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
3. Find $x$ when $P$ has been moving for 2 seconds.
\section*{LL $\_\_\_\_$}

Edexcel M3 Q5

12 marks Standard +0.8

5. A particle of mass 0.8 kg is moving along the positive $x$-axis at a speed of $5 \mathrm {~ms} ^ { - 1 }$ away from the origin $O$. When the particle is 2 metres from $O$ it becomes subject to a single force directed towards $O$. The magnitude of the force is $\frac { k } { x ^ { 2 } } \mathrm {~N}$ when the particle is $x$ metres from $O$. Given that when the particle is 4 m from $O$ its speed has been reduced to $3 \mathrm {~ms} ^ { - 1 }$,

show that $k = \frac { 128 } { 5 }$,
find the distance of the particle from $O$ when it comes to instantaneous rest. (4 marks)

Edexcel M3 2002 January Q1

8 marks Standard +0.8

A particle $P$ of mass 0.2 kg moves away from the origin along the positive $x$-axis. It moves under the action of a force directed away from the origin $O$, of magnitude $\frac{5}{x+1}$ N, where $OP = x$ metres. Given that the speed of $P$ is 5 m s$^{-1}$ when $x = 0$, find the value of $x$, to 3 significant figures, when the speed of $P$ is 15 m s$^{-1}$. [8]

Edexcel M3 Q3

8 marks Standard +0.8

A particle $P$ of mass 0.5 kg moves along a straight line. When $P$ is at a distance $x$ m from a fixed point $O$ on the line, the force acting on it is directed towards $O$ and has magnitude $\frac{8}{x}$ N. When $x = 2$, the speed of $P$ is 4 ms$^{-1}$. Find the speed of $P$ when it is 0.5 m from $O$. [8 marks]

Edexcel M3 Q4

11 marks Challenging +1.8

A particle $P$ of mass $m$ kg moves along a straight line under the action of a force of magnitude $\frac{km}{x^2}$ N, where $k$ is a constant, directed towards a fixed point $O$ on the line, where $OP = x$ m. $P$ starts from rest at $A$, at a distance $a$ m from $O$. When $OP = x$ m, the speed of $P$ is $v$ ms$^{-1}$.

Show that $v = \sqrt{\frac{2k(a-x)}{ax}}$. [6 marks]

$B$ is the point half-way between $O$ and $A$. When $k = \frac{1}{2}$ and $a = 1$, the time taken by $P$ to travel from $A$ to $B$ is $T$ seconds Assuming the result that, for $0 \leq x \leq 1$, $\int \sqrt{\frac{x}{1-x}} dx = \arcsin(\sqrt{x}) - \sqrt{x(1-x^2)} + \text{constant}$,

find the value of $T$. [5 marks]

OCR MEI Further Mechanics Major 2019 June Q10

8 marks Challenging +1.2

A particle P, of mass $m$, moves on a rough horizontal table. P is attracted towards a fixed point O on the table by a force of magnitude $\frac{kmg}{x^2}$, where $x$ is the distance OP. The coefficient of friction between P and the table is $\mu$. P is initially projected in a direction directly away from O. The velocity of P is first zero at a point A which is a distance $a$ from O.

Show that the velocity $v$ of P, when P is moving away from O, satisfies the differential equation $$\frac{\mathrm{d}}{\mathrm{d}x}(v^2) + \frac{2kg}{x^2} + 2\mu g = 0.$$ [3]
Verify that $$v^2 = 2gk\left(\frac{1}{x} - \frac{1}{a}\right) + 2\mu g(a-x).$$ [3]
Find, in terms of $k$ and $a$, the range of values of $\mu$ for which P remains at A. [2]