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Edexcel M3 Q4

11 marks Challenging +1.2

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

Show that $v = \sqrt { \frac { 2 k ( a - x ) } { a x } }$. $B$ 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 $0 \leq x \leq 1 , \int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x = \arcsin ( \sqrt { } x ) - \sqrt { } \left( x - x ^ { 2 } \right) +$ constant,
find the value of $T$. \section*{MECHANICS 3 (A) TEST PAPER 9 Page 2}

Edexcel M3 Q5

13 marks Standard +0.3

A car moves round a circular racing track of radius 100 m , which is banked at an angle of $4 ^ { \circ }$ to the horizontal.
1. Show that when its speed is $8.28 \mathrm {~ms} ^ { - 1 }$, there is no sideways force acting on the car.
  (4 marks)
2. When the speed of the car is $12.5 \mathrm {~ms} ^ { - 1 }$, find the smallest value of the coefficient of friction between the car and the track which will prevent side-slip.
3. The diagram shows a particle $P$ of mass $m \mathrm {~kg}$ moving on the inner surface of a smooth fixed hemispherical bowl of radius $r \mathrm {~m}$ which is fixed with its axis vertical. $P$ moves at a constant speed in a horizontal circle, at a depth $h \mathrm {~m}$ below the top of the bowl. \includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-2_254_431_786_1528}
4. Show that the force $R$ exerted on $P$ by the bowl has magnitude $\frac { m g r } { h } \mathrm {~N}$.
5. Find, in terms of $g , h$ and $r$, the constant speed of $P$.
The bowl is now inverted and $P$ moves on the smooth outer surface at a height $h$ above the plane face under the action of a force of magnitude $m g$ applied tangentially as shown. The reaction of the \includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-2_209_355_1224_1636}
surface of the sphere on $P$ now has magnitude $S \mathrm {~N}$.
Given that $r = 2 h$, prove that $S < \frac { 1 } { 6 } R$.

Edexcel M3 Q7

15 marks Challenging +1.8

7. A particle $P$ of mass $m \mathrm {~kg}$ is fixed to one end of a light elastic string of modulus $m g \mathrm {~N}$ and natural length $l \mathrm {~m}$. The other end of the string is attached to a fixed point $O$ on a rough horizontal table. Initially $P$ is at rest in limiting equilibrium on the table at the point $X$ where $O X = \frac { 5 l } { 4 } \mathrm {~m}$.

Find the coefficient of friction between $P$ and the table. $P$ is now given a small displacement $x \mathrm {~m}$ horizontally along $O X$, away from $O$. While $P$ is in motion, the frictional resistance remains constant at its limiting value.
Show that as long as the string remains taut, $P$ performs simple harmonic motion with $X$ as the centre. If $P$ is held at the point where the extension in the string is $l m$ and then released,
show that the string becomes slack after a time $\left( \frac { \pi } { 2 } + \arcsin \left( \frac { 1 } { 3 } \right) \right) \sqrt { \frac { l } { g } } \mathrm {~s}$.
Determine the speed of $P$ when it reaches $O$.

Edexcel M3 Q1

6 marks Standard +0.3

A cyclist travels on a banked track inclined at $8 ^ { \circ }$ to the horizontal. He moves in a horizontal circle of radius 10 m at a constant speed of $v \mathrm {~ms} ^ { - 1 }$. If there is no sideways frictional force on the cycle, calculate the value of $v$.
The figure shows a particle $P$, of mass 0.8 kg , attached to the ends of two light elastic strings. $A P$ has natural length 20 cm and modulus of elasticity $\lambda \mathrm { N } . B P$ has natural length 20 cm and modulus of of elasticity $\mu \mathrm { N } . A$ and $B$ are fixed to points on the same horizontal \includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-1_211_287_726_1704}
level so that $A B = 50 \mathrm {~cm}$. When $P$ is suspended in equilibrium, $A P =$ 30 cm and $B P = 40 \mathrm {~cm}$. Calculate the values of $\lambda$ and $\mu$.
Suraiya, whose mass is $m \mathrm {~kg}$, takes a running jump into a swimming pool so that she begins to swim in a straight line with speed $0.2 \mathrm {~ms} ^ { - 1 }$. She continues to move in the same straight line, the only force acting on her being a resistance of magnitude $m v ^ { 2 } \sin \left( \frac { t } { 100 } \right) \mathrm { N }$, where $v \mathrm {~ms} ^ { - 1 }$ is her speed at time $t$ seconds after entering the pool and $0 \leq t \leq 50 \pi$.
1. Find an expression for $v$ in terms of $t$.
2. Calculate her greatest and least speeds during her motion.
3. A uniform lamina is in the shape of the region enclosed by the coordinate axes and the curve with equation $y = 1 + \cos x$, as shown.
4. Show by integration that the centre of mass of the lamina is at a distance $\frac { \pi ^ { 2 } - 4 } { 2 \pi }$ from the $y$-axis.
  (9 marks) \includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-1_314_410_1677_1512}
Given that the centre of mass is at a distance 0.75 units from the $x$-axis, and that $P$ is the point $( 0,2 )$ and $O$ is the origin $( 0,0 )$,
find, to the nearest degree, the angle between the line $O P$ and the vertical when the lamina is freely suspended from $P$.

Edexcel M3 Q5

12 marks Challenging +1.8

5. A particle $P$, of mass 0.5 kg , rests on the surface of a rough horizontal table. The coefficient of friction between $P$ and the table is $0.5 . P$ is connected to a particle $Q$, of mass 0.2 kg , by a light inextensible string passing through \includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-1_280_519_2367_1499}
[0pt] [Turn over ... \section*{MECHANICS 3 (A) TEST PAPER 10 Page 2}

continued ...
a small smooth hole at a point $O$ on the table, such that the distance $O Q$ is $0.4 \mathrm {~m} . Q$ moves in a horizontal circle while $P$ remains in limiting equilibrium.
1. Calculate the angle $\theta$ which $O Q$ makes with the vertical.
2. Show that the speed of $Q$ is $1.33 \mathrm {~ms} ^ { - 1 }$.
The motion is altered so that $Q$ hangs at rest below $O$ and $P$ moves in a horizontal circle on the table with speed $0.84 \mathrm {~ms} ^ { - 1 }$, at a constant distance $r \mathrm {~m}$ from $O$ but tending to slip away from $O$.
Find the value of $r$.

Edexcel M3 Q6

12 marks Standard +0.8

6. The figure shows a swing consisting of two identical vertical light springs attached symmetrically to a light horizontal cross-bar and supported from a strong fixed horizontal beam. When a mass of 24 kg is placed at the mid- \includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-2_200_318_923_1668}
point of the cross-bar, both springs extend by 30 cm to the position $A$, as shown. Each spring has natural length $l \mathrm {~m}$ and modulus of elasticity $\lambda \mathrm { N }$.

Show that $\lambda = 392 l$. The 24 kg mass is left on the bar and the bar is then displaced downwards by a further 20 cm .
Prove that the system comprising the bar and the mass now performs simple harmonic motion with the centre of oscillation at the level $A$.
Calculate the number of oscillations made per second in this motion.
Find the maximum acceleration which the mass experiences during the motion.

Edexcel M3 Q7

14 marks Challenging +1.8

7. A particle $P$ of mass $m \mathrm {~kg}$ is attached to points $C$ and $D$ on the same horizontal level by means of two light inextensible strings $C P$ and $D P$, both of length $40 \mathrm {~cm} . P$ is projected with speed $u \mathrm {~ms} ^ { - 1 }$ so as to move in a vertical circle in a plane perpendicular to $C D$, so that angle $P C D =$ angle $P D C = \theta$ throughout the motion. \includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-2_335_405_1775_1572} If $u$ is just large enough for the strings to remain taut as $P$ describes this circular path,

show that $u ^ { 2 } = 2 g \sin \theta$. The string $D P$ breaks when $P$ is at its lowest point. $P$ then immediately starts to move in a horizontal circle on the end of the string $C P$.
Prove that $\tan \theta = \frac { 1 } { 5 } \sqrt { 5 }$.

OCR M3 Q4

11 marks Standard +0.8

4 \includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-02_283_711_1754_722} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, $A$ has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is moving perpendicular to the line of centres, and $B$ has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

OCR M3 Q5

10 marks Challenging +1.2

5 \includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_462_1109_283_569} Two uniform rods $A B$ and $B C$ have weights 64 N and 40 N respectively. The rods are freely jointed to each other at $B$. The rod $A B$ is freely jointed to a fixed point on horizontal ground at $A$ and the rod $B C$ rests against a vertical wall at $C$. The rod $B C$ is 1.8 m long and is horizontal. A particle of weight 9 N is attached to the rod $B C$ at the point 0.4 m from $C$. The point $A$ is 1.2 m below the level of $B C$ and 3.8 m from the wall (see diagram). The system is in equilibrium.

Show that the magnitude of the frictional force at $C$ is 27 N .
Calculate the horizontal and vertical components of the force exerted on $A B$ at $B$.
Given that friction is limiting at $C$, find the coefficient of friction between the $\operatorname { rod } B C$ and the wall.

OCR M3 Q6

14 marks Standard +0.8

6 \includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_598_839_1480_706} One end of a light inextensible string of length 0.5 m is attached to a fixed point $O$. A particle $P$ of mass 0.3 kg is attached to the other end of the string. With the string taut and at an angle of $60 ^ { \circ }$ to the upward vertical, $P$ is projected with speed $2 \mathrm {~ms} ^ { - 1 }$ (see diagram). $P$ begins to move without air resistance in a vertical circle with centre $O$. When the string makes an angle $\theta$ with the upward vertical, the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $v ^ { 2 } = 8.9 - 9.8 \cos \theta$.
Find the tension in the string in terms of $\theta$.
$P$ does not move in a complete circle. Calculate the angle through which $O P$ turns before $P$ leaves the circular path.

OCR M3 2006 January Q1

7 marks Standard +0.3

1 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_246_693_278_731} A particle $P$ of mass 0.4 kg moving in a straight line has speed $8.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. An impulse applied to $P$ deflects it through $45 ^ { \circ }$ and reduces its speed to $5.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see diagram). Calculate the magnitude and direction of the impulse exerted on $P$. $2 \quad O$ is a fixed point on a horizontal straight line. A particle $P$ of mass 0.5 kg is released from rest at $O$. At time $t$ seconds after release the only force acting on $P$ has magnitude $\left( 1 + k t ^ { 2 } \right) \mathrm { N }$ and acts horizontally and away from $O$ along the line, where $k$ is a positive constant.

Find the speed of $P$ in terms of $k$ and $t$.
Given that $P$ is 2 m from $O$ when $t = 1$, find the value of $k$ and the time taken by $P$ to travel 20 m from $O$.

OCR M3 2006 January Q3

8 marks Standard +0.3

3 A light elastic string has natural length 3 m . One end is attached to a fixed point $O$ and the other end is attached to a particle of mass 1.6 kg . The particle is released from rest in a position 5 m vertically below $O$. Air resistance may be neglected.

Given that in the subsequent motion the particle just reaches $O$, show that the modulus of elasticity of the string is 117.6 N .
Calculate the speed of the particle when it is 4.5 m below $O$.

OCR M3 2006 January Q4

10 marks Challenging +1.2

4 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_283_711_1754_722} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, $A$ has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is moving perpendicular to the line of centres, and $B$ has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

OCR M3 2006 January Q5

11 marks Challenging +1.2

5 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_462_1109_283_569} Two uniform rods $A B$ and $B C$ have weights 64 N and 40 N respectively. The rods are freely jointed to each other at $B$. The rod $A B$ is freely jointed to a fixed point on horizontal ground at $A$ and the rod $B C$ rests against a vertical wall at $C$. The rod $B C$ is 1.8 m long and is horizontal. A particle of weight 9 N is attached to the rod $B C$ at the point 0.4 m from $C$. The point $A$ is 1.2 m below the level of $B C$ and 3.8 m from the wall (see diagram). The system is in equilibrium.

Show that the magnitude of the frictional force at $C$ is 27 N .
Calculate the horizontal and vertical components of the force exerted on $A B$ at $B$.
Given that friction is limiting at $C$, find the coefficient of friction between the $\operatorname { rod } B C$ and the wall.

OCR M3 2006 January Q6

12 marks Standard +0.3

6 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_598_839_1480_706} One end of a light inextensible string of length 0.5 m is attached to a fixed point $O$. A particle $P$ of mass 0.3 kg is attached to the other end of the string. With the string taut and at an angle of $60 ^ { \circ }$ to the upward vertical, $P$ is projected with speed $2 \mathrm {~ms} ^ { - 1 }$ (see diagram). $P$ begins to move without air resistance in a vertical circle with centre $O$. When the string makes an angle $\theta$ with the upward vertical, the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $v ^ { 2 } = 8.9 - 9.8 \cos \theta$.
Find the tension in the string in terms of $\theta$.
$P$ does not move in a complete circle. Calculate the angle through which $O P$ turns before $P$ leaves the circular path.

OCR M3 2006 January Q7

16 marks Challenging +1.2

7 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-4_122_1009_265_571} As shown in the diagram, $A$ and $B$ are fixed points on a smooth horizontal table, where $A B = 3 \mathrm {~m}$. A particle $Q$ of mass 1.2 kg is attached to $A$ by a light elastic string of natural length 1 m and modulus of elasticity $180 \mathrm {~N} . Q$ is attached to $B$ by a light elastic string of natural length 1.2 m and modulus of elasticity 360 N .

Verify that when $Q$ is in equilibrium $B Q = 1.5 \mathrm {~m}$. $Q$ is projected towards $B$ from the equilibrium position with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Subsequently $Q$ oscillates with simple harmonic motion.
Show that the period of the motion is 0.314 s approximately.
Show that $u \leqslant 6$.
Given that $u = 6$, find the time taken for $Q$ to move from the equilibrium position to a position 1.3 m from $A$ for the first time.

By taking moments for $A B$ about $A$, find the vertical component of the force on $A B$ at $B$. Hence or otherwise find the magnitude of the vertical component of the contact force on $B C$ at $C$. [3]
Calculate the magnitude of the frictional force on $B C$ at $C$ and state its direction.
Calculate the value of the coefficient of friction at $C$.

OCR M3 2007 January Q4

13 marks Standard +0.3

4 A particle $P$ of mass 0.2 kg is suspended from a fixed point $O$ by a light elastic string of natural length 0.7 m and modulus of elasticity $3.5 \mathrm {~N} . P$ is at the equilibrium position when it is projected vertically downwards with speed $1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t \mathrm {~s}$ after being set in motion $P$ is $x \mathrm {~m}$ below the equilibrium position and has velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that the equilibrium position of $P$ is 1.092 m below $O$.
Prove that $P$ moves with simple harmonic motion, and calculate the amplitude.
Calculate $x$ and $v$ when $t = 0.4$.

OCR M3 2007 January Q5

12 marks Standard +0.8

5 The pilot of a hot air balloon keeps it at a fixed altitude by dropping sand from the balloon. Each grain of sand has mass $m \mathrm {~kg}$ and is released from rest. When a grain has fallen a distance $x \mathrm {~m}$, it has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Each grain falls vertically and the only forces acting on it are its weight and air resistance of magnitude $m k v ^ { 2 } \mathrm {~N}$, where $k$ is a positive constant.

Show that $\left( \frac { v } { g - k v ^ { 2 } } \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1$.
Find $v ^ { 2 }$ in terms of $k , g$ and $x$. Hence show that, as $x$ becomes large, the limiting value of $v$ is $\sqrt { \frac { g } { k } }$.
Given that the altitude of the balloon is 300 m and that each grain strikes the ground at $90 \%$ of its limiting velocity, find $k$.

OCR M3 2007 January Q6

12 marks Challenging +1.2

6 \includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-3_446_821_1007_664} Two uniform smooth spheres $A$ and $B$ of equal radius are moving on a horizontal surface when they collide. $A$ has mass 0.4 kg , and $B$ has mass $m \mathrm {~kg}$. Immediately before the collision, $A$ is moving with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an acute angle $\theta$ to the line of centres, and $B$ is moving with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $30 ^ { \circ }$ to the line of centres. Immediately after the collision $A$ is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $45 ^ { \circ }$ to the line of centres, and $B$ is moving with speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ perpendicular to the line of centres (see diagram).

Find $u$.
Given that $\theta = 88.1 ^ { \circ }$ correct to 1 decimal place, calculate the approximate values of $v$ and $m$.
The coefficient of restitution is 0.75 . Show that the exact value of $\theta$ is a root of the equation $8 \sin \theta - 6 \cos \theta = 9 \cos 30 ^ { \circ }$.

OCR M3 2007 January Q7

13 marks Challenging +1.2

7 \includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-4_721_691_269_726} The diagram shows a particle $P$ of mass 0.5 kg attached to the highest point $A$ of a fixed smooth sphere by a light elastic string. The sphere has centre $O$ and radius 1.2 m . The string has natural length 0.6 m and modulus of elasticity $6.86 \mathrm {~N} . P$ is released from rest at a point on the surface of the sphere where the acute angle $A O P$ is at least 0.5 radians.

(a) For the case angle $A O P = \alpha , P$ remains at rest. Show that $\sin \alpha = 2.8 \alpha - 1.4$.
(b) Use the iterative formula $$\alpha _ { n + 1 } = \frac { \sin \alpha _ { n } } { 2.8 } + 0.5 ,$$ with $\alpha _ { 1 } = 0.8$, to find $\alpha$ correct to 2 significant figures.
Given instead that angle $A O P = 0.5$ radians when $P$ is released, find the speed of $P$ when angle $A O P = 0.8$ radians, given that $P$ is at all times in contact with the surface of the sphere. State whether the speed of $P$ is increasing or decreasing when angle $A O P = 0.8$ radians.

OCR M3 2008 January Q1

6 marks Moderate -0.8

1 A smooth horizontal surface lies in the $x - y$ plane. A particle $P$ of mass 0.5 kg is moving on the surface with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the $x$-direction when it is struck by a horizontal blow whose impulse has components - 3.5 N s and 2.4 N s in the $x$-direction and $y$-direction respectively.

Find the components in the $x$-direction and the $y$-direction of the velocity of $P$ immediately after the blow. Hence show that the speed of $P$ immediately after the blow is $5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. $P$ is struck by a second horizontal blow whose impulse is $\mathbf { I }$.
Given that $P$ 's direction of motion immediately after this blow is parallel to the $x$-axis, write down the component of $\mathbf { I }$ in the $y$-direction.

OCR M3 2008 January Q2

9 marks Standard +0.8

2 \includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-2_515_1065_861_541} Two uniform rods $A B$ and $B C$, each of length 2 m , are freely jointed at $B$. The weights of the rods are $W \mathrm {~N}$ and 50 N respectively. The end $A$ of $A B$ is hinged at a fixed point. The rods $A B$ and $B C$ make angles $\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)$ and $\beta$ respectively with the downward vertical, and are held in equilibrium in a vertical plane by a horizontal force of magnitude 75 N acting at $C$ (see diagram).

By taking moments about $B$ for $B C$, show that $\tan \beta = 3$.
Write down the horizontal and vertical components of the force acting on $A B$ at $B$.
Find the value of $W$.

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