Questions M4 - A-Level Maths

OCR M4 2014 June Q1

1 Alan is running in a straight line on a bearing of $090 ^ { \circ }$ at a constant speed of $4 \mathrm {~ms} ^ { - 1 }$. Ben sees Alan when they are 50 m apart and Alan is on a bearing of $060 ^ { \circ }$ from Ben. Ben sets off immediately to intercept Alan by running at a constant speed of $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Calculate the bearing on which Ben should run to intercept Alan.
Calculate the magnitude of the velocity of Ben relative to Alan and find the time it takes, from the moment Ben sees Alan, for Ben to intercept Alan.

OCR M4 2014 June Q2

2 A uniform solid circular cone has mass $M$ and base radius $R$.

Show by integration that the moment of inertia of the cone about its axis of symmetry is $\frac { 3 } { 10 } M R ^ { 2 }$. (You may assume the standard formula $\frac { 1 } { 2 } m r ^ { 2 }$ for the moment of inertia of a uniform disc about its axis and that the volume of a cone is $\frac { 1 } { 3 } \pi r ^ { 2 } h$.) The axis of symmetry of the cone is fixed vertically and the cone is rotating about its axis at an angular speed of $6 \mathrm { rad } \mathrm { s } ^ { - 1 }$. A frictional couple of constant moment 0.027 Nm is applied to the cone bringing it to rest. Given that the mass of the cone is 2 kg and its base radius is 0.3 m , find
the constant angular deceleration of the cone,
the time taken for the cone to come to rest from the instant that the couple is applied.

OCR M4 2014 June Q3

3 The region bounded by the $y$-axis and the curves $y = \sin 2 x$ and $y = \sqrt { 2 } \cos x$ for $0 \leqslant x \leqslant \frac { 1 } { 4 } \pi$ is occupied by a uniform lamina. Find the exact value of the $x$-coordinate of the centre of mass of the lamina.

OCR M4 2014 June Q4

4 A uniform square lamina has mass $m$ and sides of length $2 a$.

Calculate the moment of inertia of the lamina about an axis through one of its corners perpendicular to its plane.
\includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-3_693_640_434_715} The uniform square lamina has centre $C$ and is free to rotate in a vertical plane about a fixed horizontal axis passing through one of its corners $A$. The lamina is initially held such that $A C$ is vertical with $C$ above $A$. The lamina is slightly disturbed from rest from this initial position. When $A C$ makes an angle $\theta$ with the upward vertical, the force exerted by the axis on the lamina has components $X$ parallel to $A C$ and $Y$ perpendicular to $A C$ (see diagram).
Show that the angular speed, $\omega$, of the lamina satisfies $a \omega ^ { 2 } = \frac { 3 } { 4 } g \sqrt { 2 } ( 1 - \cos \theta )$.
Find $X$ and $Y$ in terms of $m , g$ and $\theta$. \section*{Question 5 begins on page 4.}
\includegraphics[max width=\textwidth, alt={}]{639c658e-0aca-4161-9e77-0f4c494b0b55-4_767_337_248_863}
A pendulum consists of a uniform rod $A B$ of length $4 a$ and mass $4 m$ and a spherical shell of radius $a$, mass $m$ and centre $C$. The end $B$ of the rod is rigidly attached to a point on the surface of the shell in such a way that $A B C$ is a straight line. The pendulum is initially at rest with $B$ vertically below $A$ and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through $A$ (see diagram).
Show that the moment of inertia of the pendulum about the axis of rotation is $47 m a ^ { 2 }$. A particle of mass $m$ is moving horizontally in the plane in which the pendulum is free to rotate. The particle has speed $\sqrt { k g a }$, where $k$ is a positive constant, and strikes the rod at a distance $3 a$ from $A$. In the subsequent motion the particle adheres to the rod and the combined rigid body $P$ starts to rotate.
Show that the initial angular speed of $P$ is $\frac { 3 } { 56 } \sqrt { \frac { k g } { a } }$.
For the case $k = 4$, find the angle that $P$ has turned through when $P$ first comes to instantaneous rest.
Find the least value of $k$ such that the rod reaches the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-5_437_903_269_573} A uniform rod $A B$ has mass $m$ and length $2 a$. The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through $A$. One end of a light elastic string of natural length $a$ and modulus of elasticity $\sqrt { 3 } m g$ is attached to $A$. The string passes over a small smooth fixed pulley $C$, where $A C$ is horizontal and $A C = a$. The other end of the string is attached to the rod at its mid-point $D$. The rod makes an angle $\theta$ below the horizontal (see diagram).
Taking $A$ as the reference level for gravitational potential energy, show that the total potential energy $V$ of the system is given by $$V = m g a ( \sqrt { 3 } - \sin \theta - \sqrt { 3 } \cos \theta ) .$$
Show that $\theta = \frac { 1 } { 6 } \pi$ is a position of stable equilibrium for the system. The system is making small oscillations about the equilibrium position.
By differentiating the energy equation with respect to time, show that $$\frac { 4 } { 3 } a \ddot { \theta } = g ( \cos \theta - \sqrt { 3 } \sin \theta ) .$$
Using the substitution $\theta = \phi + \frac { 1 } { 6 } \pi$, show that the motion is approximately simple harmonic, and find the approximate period of the oscillations. \section*{END OF QUESTION PAPER}

OCR M4 2015 June Q1

1 A turntable is rotating at $3 \mathrm { rad } \mathrm { s } ^ { - 1 }$. The turntable is then accelerated so that after 4 revolutions it is rotating at $12.4 \mathrm { rad } \mathrm { s } ^ { - 1 }$. Assuming that the angular acceleration of the turntable is constant,

find the angular acceleration,
find the time taken to increase its angular speed from $3 \mathrm { rad } \mathrm { s } ^ { - 1 }$ to $12.4 \mathrm { rad } \mathrm { s } ^ { - 1 }$.

OCR M4 2015 June Q2

2 The region bounded by the $x$-axis, the lines $x = 1$ and $x = 2$, and the curve $y = k x ^ { 2 }$, where $k$ is a positive constant, is occupied by a uniform lamina.

Find the exact $x$-coordinate of the centre of mass of the lamina.
Given that the $x$ - and $y$-coordinates of the centre of mass of the lamina are equal, find the exact value of $k$.

OCR M4 2015 June Q3

3 Two planes, $A$ and $B$, flying at the same altitude, are participating in an air show. Initially the planes are 400 m apart and plane $B$ is on a bearing of $130 ^ { \circ }$ from plane $A$. Plane $A$ is moving due south with a constant speed of $75 \mathrm {~ms} ^ { - 1 }$. Plane $B$ is moving at a constant speed of $40 \mathrm {~ms} ^ { - 1 }$ and has set a course to get as close as possible to $A$.

Find the bearing of the course set by $B$ and the shortest distance between the two planes in the subsequent motion.
Find the total distance travelled by $A$ and $B$ from the instant when they are initially 400 m apart to the point of their closest approach.

OCR M4 2015 June Q4

4

Write down the moment of inertia of a uniform circular disc of mass $m$ and radius $2 a$ about a diameter. A uniform solid cylinder has mass $M$, radius $2 r$ and height $h$.
Show by integration, and using the result from part (i), that the moment of inertia of the cylinder about a diameter of an end face is $$M \left( r ^ { 2 } + \frac { 1 } { 3 } h ^ { 2 } \right)$$ and hence find the moment of inertia of the cylinder about a diameter through the centre of the cylinder.
\includegraphics[max width=\textwidth, alt={}, center]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-3_919_897_260_591} A smooth circular wire hoop, with centre $O$ and radius $r$, is fixed in a vertical plane. The highest point on the wire is $H$. A small bead $B$ of mass $m$ is free to move along the wire. A light inextensible string of length $a$, where $a > 2 r$, has one end attached to the bead. The other end of the string passes over a small smooth pulley at $H$ and carries at its end a particle $P$ of mass $\lambda m$, where $\lambda$ is a positive constant. The part of the string $H P$ is vertical and the part of the string $B H$ makes an angle $\theta$ radians with the downward vertical where $0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$ (see diagram). You may assume that $P$ remains above the lowest point of the wire.

OCR M4 2015 June Q6

6 A pendulum consists of a uniform rod $A B$ of length $2 a$ and mass $2 m$ and a particle of mass $m$ that is attached to the end $B$. The pendulum can rotate in a vertical plane about a smooth fixed horizontal axis passing through $A$.

Show that the moment of inertia of this pendulum about the axis of rotation is $\frac { 20 } { 3 } m a ^ { 2 }$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_572_86_852_575} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_582_456_842_1050} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The pendulum is initially held with $B$ vertically above $A$ (see Fig.1) and it is slightly disturbed from this position. When the angle between the pendulum and the upward vertical is $\theta$ radians the pendulum has angular speed $\omega \mathrm { rads } ^ { - 1 }$ (see Fig. 2).
Show that $$\omega ^ { 2 } = \frac { 6 g } { 5 a } ( 1 - \cos \theta ) .$$
Find the angular acceleration of the pendulum in terms of $g , a$ and $\theta$. At an instant when $\theta = \frac { 1 } { 3 } \pi$, the force acting on the pendulum at $A$ has magnitude $F$.
Find $F$ in terms of $m$ and $g$. It is given that $a = 0.735 \mathrm {~m}$.
Show that the time taken for the pendulum to move from the position $\theta = \frac { 1 } { 6 } \pi$ to the position $\theta = \frac { 1 } { 3 } \pi$ is given by $$k \int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta ,$$ stating the value of the constant $k$. Hence find the time taken for the pendulum to rotate between these two points. (You may quote an appropriate result given in the List of Formulae (MF1).) \section*{END OF QUESTION PAPER}

OCR M4 2016 June Q1

1 A uniform square lamina, of mass 5 kg and side 0.2 m , is rotating about a fixed vertical axis that is perpendicular to the lamina and that passes through its centre. A couple of constant moment 0.06 Nm is applied to the lamina. The lamina turns through an angle of 155 radians while its angular speed increases from $8 \mathrm { rads } ^ { - 1 }$ to $\omega \mathrm { rads } ^ { - 1 }$. Find $\omega$.

OCR M4 2016 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{27b790da-800f-4f5e-8f63-d52159efb48e-2_959_1166_609_450} Boat $A$ is travelling with constant speed $7.9 \mathrm {~ms} ^ { - 1 }$ on a course with bearing $035 ^ { \circ }$. Boat $B$ is travelling with constant speed $10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a course with bearing $330 ^ { \circ }$. At one instant, the boats are 1500 m apart with $B$ on a bearing of $125 ^ { \circ }$ from $A$ (see diagram).

Find the magnitude and the bearing of the velocity of $B$ relative to $A$.
Find the shortest distance between $A$ and $B$ in the subsequent motion.
Find the time taken from the instant when $A$ and $B$ are 1500 m apart to the instant when $A$ and $B$ are at the point of closest approach.
\includegraphics[max width=\textwidth, alt={}, center]{27b790da-800f-4f5e-8f63-d52159efb48e-3_1057_1047_248_511} Two uniform rods $A B$ and $B C$, each of length $a$ and mass $m$, are rigidly joined together so that $A B$ is perpendicular to $B C$. The rod $A B$ is freely hinged to a fixed point at $A$. The rods can rotate in a vertical plane about a smooth fixed horizontal axis through $A$. One end of a light elastic string of natural length $a$ and modulus of elasticity $\lambda m g$ is attached to $B$. The other end of the string is attached to a fixed point $D$ vertically above $A$, where $A D = a$. The string $B D$ makes an angle $\theta$ radians with the downward vertical (see diagram).
Taking $D$ as the reference level for gravitational potential energy, show that the total potential energy $V$ of the system is given by $$\mathrm { V } = \frac { 1 } { 2 } \mathrm { mga } ( \sin 2 \theta - 3 \cos 2 \theta ) + \frac { 1 } { 2 } \lambda \mathrm { mga } ( 2 \cos \theta - 1 ) ^ { 2 } - 2 \mathrm { mga } .$$
Given that $\theta = \frac { 1 } { 4 } \pi$ is a position of equilibrium, find the exact value of $\lambda$.
Find $\frac { d ^ { 2 } V } { d \theta ^ { 2 } }$ and hence determine whether the position of equilibrium at $\theta = \frac { 1 } { 4 } \pi$ is stable or unstable.

OCR M4 2016 June Q4

4 The region bounded by the curve $\mathrm { y } = 2 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { x } }$ for $0 \leqslant x \leqslant 2$, the $x$-axis, the $y$-axis and the line $x = 2$, is occupied by a uniform lamina.

Find the exact value of the $y$-coordinate of the centre of mass of the lamina. As shown in the diagram below, a uniform lamina occupies the closed region bounded by the $x$-axis, the $y$-axis and the curve $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ where $$f ( x ) = \begin{cases} 2 \mathrm { e } ^ { \frac { 1 } { 2 } x } & 0 \leqslant x \leqslant 2
\frac { 2 } { 3 } ( 5 - x ) \mathrm { e } & 2 \leqslant x \leqslant 5 . \end{cases}$$ \includegraphics[max width=\textwidth, alt={}, center]{27b790da-800f-4f5e-8f63-d52159efb48e-4_863_1179_762_443}
Find the exact value of the $x$-coordinate of the centre of mass of the lamina.

OCR M4 2016 June Q5

5 A uniform rod $A B$ has mass $2 m$ and length 4a.

Show by integration that the moment of inertia of the rod about an axis perpendicular to the rod through $A$ is $\frac { 32 } { 3 } \mathrm { ma } ^ { 2 }$ The rod is initially at rest with $B$ vertically below $A$ and it is free to rotate in a vertical plane about a smooth fixed horizontal axis through $A$. A particle of mass $m$ is moving horizontally in the plane in which the rod is free to rotate. The particle has speed $v$, and strikes the rod at $B$. In the subsequent motion the particle adheres to the rod and the combined rigid body $Q$, consisting of the rod and the particle, starts to rotate.
Find, in terms of $v$ and $a$, the initial angular speed of $Q$. At time $t$ seconds the angle between $Q$ and the downward vertical is $\theta$ radians.
Show that $\dot { \theta } ^ { 2 } = \mathrm { k } \frac { \mathrm { g } } { \mathrm { a } } ( \cos \theta - 1 ) + \frac { 9 \mathrm { v } ^ { 2 } } { 400 \mathrm { a } ^ { 2 } }$, stating the value of the constant $k$.
Find, in terms of $a$ and $g$, the set of values of $v ^ { 2 }$ for which $Q$ makes complete revolutions. When $Q$ is horizontal, the force exerted by the axis on $Q$ has vertically upwards component $R$.
Find $R$ in terms of $m$ and $g$.
\includegraphics[max width=\textwidth, alt={}, center]{27b790da-800f-4f5e-8f63-d52159efb48e-6_844_509_248_778} A compound pendulum consists of a uniform rod $A B$ of length 1 m and mass 3 kg , a particle of mass 1 kg attached to the rod at $A$ and a circular disc of radius $\frac { 1 } { 3 } \mathrm {~m}$, mass 6 kg and centre $C$. The end $B$ of the rod is rigidly attached to a point on the circumference of the disc in such a way that $A B C$ is a straight line. The pendulum is initially at rest with $B$ vertically below $A$ and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point $P$ on the rod where $\mathrm { AP } = \mathrm { xm }$ and $\mathrm { x } < \frac { 1 } { 2 }$ (see diagram).
Show that the moment of inertia of the pendulum about the axis of rotation is $\left( 10 x ^ { 2 } - 19 x + 12 \right) \mathrm { kg } \mathrm { m } ^ { 2 }$. The pendulum is making small oscillations about the equilibrium position, such that at time $t$ seconds the angular displacement that the pendulum makes with the downward vertical is $\theta$ radians.
Find the angular acceleration of the pendulum, in terms of $x , g$ and $\theta$.
Show that the motion is approximately simple harmonic, and show that the approximate period of oscillations, in seconds, is given by $2 \pi \sqrt { \frac { 20 x ^ { 2 } - 38 x + 24 } { ( 19 - 20 x ) g } }$.
Hence find the value of $x$ for which the approximate period of oscillations is least.

OCR M4 2017 June Q1

1 A uniform rod with centre $C$ has mass $2 M$ and length 4a. The rod is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through a point $A$ on the rod, where $A C = k a$ and $0 < k < 2$. The rod is making small oscillations about the equilibrium position with period $T$.

Show that $T = 2 \pi \sqrt { \frac { a } { 3 g } \left( \frac { 4 + 3 k ^ { 2 } } { k } \right) }$. (You may assume the standard formula $T = 2 \pi \sqrt { \frac { I } { m g h } }$ for the period of small oscillations of a compound pendulum.)
Hence find the value of $k ^ { 2 }$ for which the period of oscillations is least.

OCR M4 2017 June Q2

2 A ship $S$ is travelling with constant speed $5 \mathrm {~ms} ^ { - 1 }$ on a course with bearing $325 ^ { \circ }$. A second ship $T$ observes $S$ when $S$ is 9500 m from $T$ on a bearing of $060 ^ { \circ }$ from $T$. Ship $T$ sets off in pursuit, travelling with constant speed $8.5 \mathrm {~ms} ^ { - 1 }$ in a straight line.

Find the bearing of the course which $T$ should take in order to intercept $S$.
Find the distance travelled by $S$ from the moment that $T$ sets off in pursuit until the point of interception.

OCR M4 2017 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-2_439_444_1318_822} A uniform rod $A B$ has mass $m$ and length $4 a$. The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through $A$. One end of a light elastic string of natural length $a$ and modulus of elasticity $\lambda m g$ is attached to $B$. The other end of the string is attached to a small light ring which slides on a fixed smooth horizontal rail which is in the same vertical plane as the rod. The rail is a vertical distance $3 a$ above $A$. The string is always vertical and the rod makes an angle $\theta$ radians with the horizontal, where $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$ (see diagram).

Taking $A$ as the reference level for gravitational potential energy, find an expression for the total potential energy $V$ of the system, and show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 m g a \cos \theta ( 4 \lambda ( 1 + 2 \sin \theta ) - 1 ) .$$ Determine the positions of equilibrium and the nature of their stability in the cases
$\lambda > \frac { 1 } { 12 }$,
$\lambda < \frac { 1 } { 12 }$.
\includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-3_392_689_269_671} The diagram shows the curve with equation $y = \frac { 1 } { 2 } \ln x$. The region $R$, shaded in the diagram, is bounded by the curve, the $x$-axis and the line $x = 4$. A uniform solid of revolution is formed by rotating $R$ completely about the $y$-axis to form a solid of volume $V$.
Show that $V = \frac { 1 } { 4 } \pi ( 64 \ln 2 - 15 )$.
Find the exact $y$-coordinate of the centre of mass of the solid. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{57323af2-8cf3-4721-b2c8-a968264be343-4_385_741_269_646} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Fig. 1 shows part of the line $y = \frac { a } { h } x$, where $a$ and $h$ are constants. The shaded region bounded by the line, the $x$-axis and the line $x = h$ is rotated about the $x$-axis to form a uniform solid cone of base radius $a$, height $h$ and volume $\frac { 1 } { 3 } \pi a ^ { 2 } h$. The mass of the cone is $M$.
Show by integration that the moment of inertia of the cone about the $y$-axis is $\frac { 3 } { 20 } M \left( a ^ { 2 } + 4 h ^ { 2 } \right)$. (You may assume the standard formula $\frac { 1 } { 4 } m r ^ { 2 }$ for the moment of inertia of a uniform disc about a diameter.) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{57323af2-8cf3-4721-b2c8-a968264be343-4_501_556_1238_726} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A uniform solid cone has mass 3 kg , base radius 0.4 m and height 1.2 m . The cone can rotate about a fixed vertical axis passing through its centre of mass with the axis of the cone moving in a horizontal plane. The cone is rotating about this vertical axis at an angular speed of $9.6 \mathrm { rad } \mathrm { s } ^ { - 1 }$. A stationary particle of mass $m \mathrm {~kg}$ becomes attached to the vertex of the cone (see Fig. 2). The particle being attached to the cone causes the angular speed to change instantaneously from $9.6 \mathrm { rad } \mathrm { s } ^ { - 1 }$ to $7.8 \mathrm { rad } \mathrm { s } ^ { - 1 }$.
Find the value of $m$.
\includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-5_534_501_255_767} A triangular frame $A B C$ consists of three uniform rods $A B , B C$ and $C A$, rigidly joined at $A , B$ and $C$. Each rod has mass $m$ and length $2 a$. The frame is free to rotate in a vertical plane about a fixed horizontal axis passing through $A$. The frame is initially held such that the axis of symmetry through $A$ is vertical and $B C$ is below the level of $A$. The frame starts to rotate with an initial angular speed of $\omega$ and at time $t$ the angle between the axis of symmetry through $A$ and the vertical is $\theta$ (see diagram).
Show that the moment of inertia of the frame about the axis through $A$ is $6 m a ^ { 2 }$.
Show that the angular speed $\dot { \theta }$ of the frame when it has turned through an angle $\theta$ satisfies $$a \dot { \theta } ^ { 2 } = a \omega ^ { 2 } - k g \sqrt { 3 } ( 1 - \cos \theta ) ,$$ stating the exact value of the constant $k$.
Hence find, in terms of $a$ and $g$, the set of values of $\omega ^ { 2 }$ for which the frame makes complete revolutions. At an instant when $\theta = \frac { 1 } { 6 } \pi$, the force acting on the frame at $A$ has magnitude $F$.
Given that $\omega ^ { 2 } = \frac { 2 g } { a \sqrt { 3 } }$, find $F$ in terms of $m$ and $g$. \section*{END OF QUESTION PAPER}

OCR MEI M4 Q2

2 A rigid circular hoop of radius $a$ is fixed in a vertical plane. At the highest point of the hoop there is a small smooth pulley, P. A light inextensible string AB of length $\frac { 5 } { 2 } a$ is passed over the pulley. A particle of mass $m$ is attached to the string at $\mathrm { B } . \mathrm { PB }$ is vertical and angle $\mathrm { APB } = \theta$. A small smooth ring of mass $m$ is threaded onto the hoop and attached to the string at A . This situation is shown in Fig. 2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5bed3ad4-0e20-4458-a37f-655faf84c31a-02_568_549_1306_758} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

Show that $\mathrm { PB } = \frac { 5 } { 2 } a - 2 a \cos \theta$ and hence show that the potential energy of the system relative to P is $V = - m g a \left( 2 \cos ^ { 2 } \theta - 2 \cos \theta + \frac { 5 } { 2 } \right)$.
Hence find the positions of equilibrium and investigate their stability.

OCR MEI M4 2006 June Q1

1 A spherical raindrop falls through a stationary cloud. Water condenses on the raindrop and it gains mass at a rate proportional to its surface area. At time $t$ the radius of the raindrop is $r$. Initially the raindrop is at rest and $r = r _ { 0 }$. The density of the water is $\rho$.

Show that $\frac { \mathrm { d } r } { \mathrm {~d} t } = k$, where $k$ is a constant. Hence find the mass of the raindrop in terms of $r _ { 0 } , \rho , k$ and $t$.
Assuming that air resistance is negligible, find the velocity of the raindrop in terms of $r _ { 0 } , k$ and $t$.

OCR MEI M4 2006 June Q2

2 A rigid circular hoop of radius $a$ is fixed in a vertical plane. At the highest point of the hoop there is a small smooth pulley, P. A light inextensible string AB of length $\frac { 5 } { 2 } a$ is passed over the pulley. A particle of mass $m$ is attached to the string at $\mathrm { B } . \mathrm { PB }$ is vertical and angle $\mathrm { APB } = \theta$. A small smooth ring of mass $m$ is threaded onto the hoop and attached to the string at A . This situation is shown in Fig. 2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c97056a9-4156-4ecd-a80e-1a82c81ab824-2_568_549_1306_758} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

Show that $\mathrm { PB } = \frac { 5 } { 2 } a - 2 a \cos \theta$ and hence show that the potential energy of the system relative to P is $V = - m g a \left( 2 \cos ^ { 2 } \theta - 2 \cos \theta + \frac { 5 } { 2 } \right)$.
Hence find the positions of equilibrium and investigate their stability.

OCR MEI M4 2006 June Q3

3 An aeroplane is taking off from a runway. It starts from rest. The resultant force in the direction of motion has power, $P$ watts, modelled by $$P = 0.0004 m \left( 10000 v + v ^ { 3 } \right) ,$$ where $m \mathrm {~kg}$ is the mass of the aeroplane and $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity at time $t$ seconds. The displacement of the aeroplane from its starting point is $x \mathrm {~m}$. To take off successfully the aeroplane must reach a speed of $80 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ before it has travelled 900 m .

Formulate and solve a differential equation for $v$ in terms of $x$. Hence show that the aeroplane takes off successfully.
Formulate a differential equation for $v$ in terms of $t$. Solve the differential equation to show that $v = 100 \tan ( 0.04 t )$. What feature of this result casts doubt on the validity of the model?
In fact the model is only valid for $0 \leqslant t \leqslant 11$, after which the power remains constant at the value attained at $t = 11$. Will the aeroplane take off successfully?

OCR MEI M4 2006 June Q4

4 A flagpole AB of length $2 a$ is modelled as a thin rigid rod of variable mass per unit length given by $$\rho = \frac { M } { 8 a ^ { 2 } } ( 5 a - x ) ,$$ where $x$ is the distance from A and $M$ is the mass of the flagpole.

Show that the moment of inertia of the flagpole about an axis through A and perpendicular to the flagpole is $\frac { 7 } { 6 } M a ^ { 2 }$. Show also that the centre of mass of the flagpole is at a distance $\frac { 11 } { 12 } a$ from A . The flagpole is hinged to a wall at A and can rotate freely in a vertical plane. A light inextensible rope of length $2 \sqrt { 2 } a$ is attached to the end B and the other end is attached to a point on the wall a distance $2 a$ vertically above A, as shown in Fig. 4. The flagpole is initially at rest when lying vertically against the wall, and then is displaced slightly so that it falls to a horizontal position, at which point the rope becomes taut and the flagpole comes to rest. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c97056a9-4156-4ecd-a80e-1a82c81ab824-4_403_365_1174_849} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
Find an expression for the angular velocity of the flagpole when it has turned through an angle $\theta$.
Show that the vertical component of the impulse in the rope when it becomes taut is $\frac { 1 } { 12 } M \sqrt { 77 a g }$. Hence write down the horizontal component.
Find the horizontal and vertical components of the impulse that the hinge exerts on the flagpole when the rope becomes taut. Hence find the angle that this impulse makes with the horizontal.

OCR MEI M4 2007 June Q1

1 A light elastic string has one end fixed to a vertical pole at A . The string passes round a smooth horizontal peg, P , at a distance $a$ from the pole and has a smooth ring of mass $m$ attached at its other end B . The ring is threaded onto the pole below A . The ring is at a distance $y$ below the horizontal level of the peg. This situation is shown in Fig. 1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-2_462_275_557_897} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} The string has stiffness $k$ and natural length equal to the distance AP .

Express the extension of the string in terms of $y$ and $a$. Hence find the potential energy of the system relative to the level of P .
Use the potential energy to find the equilibrium position of the system, and show that it is stable.
Calculate the normal reaction exerted by the pole on the ring in the equilibrium position.

OCR MEI M4 2007 June Q2

2 A railway truck of mass $m _ { 0 }$ travels along a horizontal track. There is no driving force and the resistances to motion are negligible. The truck is being filled with coal which falls vertically into it at a mass rate $k$. The process starts as the truck passes a point O with speed $u$. After time $t$, the truck has velocity $v$ and the displacement from O is $x$.

Show that $v = \frac { m _ { 0 } u } { m _ { 0 } + k t }$ and find $x$ in terms of $m _ { 0 } , u , k$ and $t$.
Find the distance that the truck has travelled when its speed has been halved.

OCR MEI M4 2007 June Q3

3

Show, by integration, that the moment of inertia of a uniform rod of mass $m$ and length $2 a$ about an axis through its centre and perpendicular to the rod is $\frac { 1 } { 3 } m a ^ { 2 }$. A pendulum of length 1 m is made by attaching a uniform sphere of mass 2 kg and radius 0.1 m to the end of a uniform rod AB of mass 1.2 kg and length 0.8 m , as shown in Fig. 3. The centre of the sphere is collinear with A and B . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-3_442_291_717_886} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
Find the moment of inertia of the pendulum about an axis through A perpendicular to the rod. The pendulum can swing freely in a vertical plane about a fixed horizontal axis through A .
The pendulum is held with AB at an angle $\alpha$ to the downward vertical and released from rest. At time $t , \mathrm { AB }$ is at an angle $\theta$ to the vertical. Find an expression for $\dot { \theta } ^ { 2 }$ in terms of $\theta$ and $\alpha$.
Hence, or otherwise, show that, provided that $\alpha$ is small, the pendulum performs simple harmonic motion. Calculate the period.

OCR MEI M4 2007 June Q4

4 A particle of mass 2 kg starts from rest at a point O and moves in a horizontal line with velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ under the action of a force $F \mathrm {~N}$, where $F = 2 - 8 v ^ { 2 }$. The displacement of the particle from O at time $t$ seconds is $x \mathrm {~m}$.

Formulate and solve a differential equation to show that $v ^ { 2 } = \frac { 1 } { 4 } \left( 1 - \mathrm { e } ^ { - 8 x } \right)$.
Hence express $F$ in terms of $x$ and find, by integration, the work done in the first 2 m of the motion.
Formulate and solve a differential equation to show that $v = \frac { 1 } { 2 } \left( \frac { 1 - \mathrm { e } ^ { - 4 t } } { 1 + \mathrm { e } ^ { - 4 t } } \right)$.
Calculate $v$ when $t = 1$ and when $t = 2$, giving your answers to four significant figures. Hence find the impulse of the force $F$ over the interval $1 \leqslant t \leqslant 2$.

Questions M4 (303 questions)

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