Questions - OCR MEI M4 - A-Level Maths

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

\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

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 $1 \leqslant t \leqslant 2$.

OCR MEI M4 2008 June Q1

1 A rocket in deep space starts from rest and moves in a straight line. The initial mass of the rocket is $m _ { 0 }$ and the propulsion system ejects matter at a constant mass rate $k$ with constant speed $u$ relative to the rocket. At time $t$ the speed of the rocket is $v$.

Show that while mass is being ejected from the rocket, $\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k$.
Hence find an expression for $v$ in terms of $t$.
Find the speed of the rocket when its mass is $\frac { 1 } { 3 } m _ { 0 }$.

OCR MEI M4 2008 June Q2

2 A car of mass $m \mathrm {~kg}$ starts from rest at a point O and moves along a straight horizontal road. The resultant force in the direction of motion has power $P$ watts, given by $P = m \left( k ^ { 2 } - v ^ { 2 } \right)$, where $v \mathrm {~ms} ^ { - 1 }$ is the velocity of the car and $k$ is a positive constant. The displacement from O in the direction of motion is $x \mathrm {~m}$.

Show that $\left( \frac { k ^ { 2 } } { k ^ { 2 } - v ^ { 2 } } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1$, and hence find $x$ in terms of $v$ and $k$.
How far does the car travel before reaching $90 \%$ of its terminal velocity?

OCR MEI M4 2008 June Q3

3 A circular disc of radius $a \mathrm {~m}$ has mass per unit area $\rho \mathrm { kg } \mathrm { m } ^ { - 2 }$ given by $\rho = k ( a + r )$, where $r \mathrm {~m}$ is the distance from the centre and $k$ is a positive constant. The disc can rotate freely about an axis perpendicular to it and through its centre.

Show that the mass, $M \mathrm {~kg}$, of the disc is given by $M = \frac { 5 } { 3 } k \pi a ^ { 3 }$, and show that the moment of inertia, $I \mathrm {~kg} \mathrm {~m} ^ { 2 }$, about this axis is given by $I = \frac { 27 } { 50 } M a ^ { 2 }$. For the rest of this question, take $M = 64$ and $a = 0.625$.
The disc is at rest when it is given a tangential impulsive blow of 50 N s at a point on its circumference.
Find the angular speed of the disc. The disc is then accelerated by a constant couple reaching an angular speed of $30 \mathrm { rad } \mathrm { s } ^ { - 1 }$ in 20 seconds.
Calculate the magnitude of this couple. When the angular speed is $30 \mathrm { rads } ^ { - 1 }$, the couple is removed and brakes are applied to bring the disc to rest. The effect of the brakes is modelled by a resistive couple of $3 \dot { \theta } \mathrm { Nm }$, where $\dot { \theta }$ is the angular speed of the disc in $\mathrm { rad } \mathrm { s } ^ { - 1 }$.
Formulate a differential equation for $\dot { \theta }$ and hence find $\dot { \theta }$ in terms of $t$, the time in seconds from when the brakes are first applied.
By reference to your expression for $\dot { \theta }$, give a brief criticism of this model for the effect of the brakes.

OCR MEI M4 2008 June Q4

4 A uniform smooth pulley can rotate freely about its axis, which is fixed and horizontal. A light elastic string AB is attached to the pulley at the end B . The end A is attached to a fixed point such that the string is vertical and is initially at its natural length with B at the same horizontal level as the axis. In this position a particle P is attached to the highest point of the pulley. This initial position is shown in Fig. 4.1. The radius of the pulley is $a$, the mass of P is $m$ and the stiffness of the string AB is $\frac { m g } { 10 a }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_451_517_607_466} \captionsetup{labelformat=empty} \caption{Fig. 4.1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_456_451_607_1226} \captionsetup{labelformat=empty} \caption{Fig. 4.2}

\end{figure}

Fig. 4.2 shows the system with the pulley rotated through an angle $\theta$ and the string stretched. Write down the extension of the string and hence find the potential energy, $V$, of the system in this position. Show that $\frac { \mathrm { d } V } { \mathrm {~d} \theta } = m g a \left( \frac { 1 } { 10 } \theta - \sin \theta \right)$.
Hence deduce that the system has a position of unstable equilibrium at $\theta = 0$.
Explain how your expression for $V$ relies on smooth contact between the string and the pulley. Fig. 4.3 shows the graph of the function $\mathrm { f } ( \theta ) = \frac { 1 } { 10 } \theta - \sin \theta$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_538_1342_1706_404} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
\end{figure}
Use the graph to give rough estimates of three further values of $\theta$ (other than $\theta = 0$ ) which give positions of equilibrium. In each case, state with reasons whether the equilibrium is stable or unstable.
Show on a sketch the physical situation corresponding to the least value of $\theta$ you identified in part (iv). On your sketch, mark clearly the positions of P and B .
The equation $\mathrm { f } ( \theta ) = 0$ has another root at $\theta \approx - 2.9$. Explain, with justification, whether this necessarily gives a position of equilibrium.

OCR MEI M4 2009 June Q1

1 A raindrop increases in mass as it falls vertically from rest through a stationary cloud. At time $t \mathrm {~s}$ the velocity of the raindrop is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its mass is $m \mathrm {~kg}$. The rate at which the mass increases is modelled as $\frac { m g } { 2 ( v + 1 ) } \mathrm { kg } \mathrm { s } ^ { - 1 }$. Resistances to motion are neglected.

Write down the equation of motion of the raindrop. Hence show that $$\left( 1 - \frac { 1 } { v + 2 } \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1 } { 2 } g .$$
Solve this differential equation to find an expression for $t$ in terms of $v$. Calculate the time it takes for the velocity of the raindrop to reach $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Describe, with reasons, what happens to the acceleration of the raindrop for large values of $t$.

OCR MEI M4 2009 June Q2

2 A uniform rigid rod AB of mass $m$ and length $4 a$ is freely hinged at the end A to a horizontal rail. The end B is attached to a light elastic string BC of modulus $\frac { 1 } { 2 } m g$ and natural length $a$. The end C of the string is attached to a ring which is small, light and smooth. The ring can slide along the rail and is always vertically above B . The angle that AB makes below the rail is $\theta$. The system is shown in Fig. 2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9763e6c4-e372-46ef-a666-3ccb185aa5d2-2_277_707_1398_717} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

Find the potential energy, $V$, of the system when the string is stretched and show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 4 m g a \cos \theta ( 2 \sin \theta - 1 )$$
Hence find any positions of equilibrium of the system and investigate their stability.

OCR MEI M4 2009 June Q3

3 A uniform circular disc has mass $M$ and radius $a$. The centre of the disc is at point C .

Show by integration that the moment of inertia of the disc about an axis through C and perpendicular to the disc is $\frac { 1 } { 2 } M a ^ { 2 }$. The point A on the disc is at a distance $\frac { 1 } { 10 } a$ from its centre.
Show that the moment of inertia of the disc about an axis through A and perpendicular to the disc is $0.51 M a ^ { 2 }$. The disc can rotate freely in a vertical plane about an axis through A that is horizontal and perpendicular to the disc. The disc is held slightly displaced from its stable equilibrium position and is released from rest. In the motion that follows, the angle that AC makes with the downward vertical is $\theta$.
Write down the equation of motion for the disc. Assuming $\theta$ remains sufficiently small throughout the motion, show that the disc performs approximate simple harmonic motion and determine the period of the motion. A particle of mass $m$ is attached at a point P on the circumference of the disc, so that the centre of mass of the system is now at A .
Sketch the position of P in relation to A and C . Find $m$ in terms of $M$ and show that the moment of inertia of the system about the axis through A and perpendicular to the disc is $0.6 M a ^ { 2 }$. The system now rotates at a constant angular speed $\omega$ about the axis through A .
Find the kinetic energy of the system. Hence find the magnitude of the constant resistive couple needed to bring the system to rest in $n$ revolutions.

OCR MEI M4 2009 June Q4

4 A parachutist of mass 90 kg falls vertically from rest. The forces acting on her are her weight and resistance to motion $R \mathrm {~N}$. At time $t \mathrm {~s}$ the velocity of the parachutist is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the distance she has fallen is $x \mathrm {~m}$. While the parachutist is in free-fall (i.e. before the parachute is opened), the resistance is modelled as $R = k v ^ { 2 }$, where $k$ is a constant. The terminal velocity of the parachutist in free-fall is $60 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $k = \frac { g } { 40 }$.
Show that $v ^ { 2 } = 3600 \left( 1 - \mathrm { e } ^ { - \frac { g x } { 1800 } } \right)$. When she has fallen 1800 m , she opens her parachute.
Calculate, by integration, the work done against the resistance before she opens her parachute. Verify that this is equal to the loss in mechanical energy of the parachutist. As the parachute opens, the resistance instantly changes and is now modelled as $R = 90 v$.
Calculate her velocity just before opening the parachute, correct to four decimal places.
Formulate and solve a differential equation to calculate the time it takes after opening the parachute to reduce her velocity to $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

OCR MEI M4 2010 June Q1

1 At time $t$ a rocket has mass $m$ and is moving vertically upwards with velocity $v$. The propulsion system ejects matter at a constant speed $u$ relative to the rocket. The only additional force acting on the rocket is its weight.

Derive the differential equation $m \frac { \mathrm {~d} v } { \mathrm {~d} t } + u \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g$. The rocket has initial mass $m _ { 0 }$ of which $75 \%$ is fuel. It is launched from rest. Matter is ejected at a constant mass rate $k$.
Assuming that the acceleration due to gravity is constant, find the speed of the rocket at the instant when all the fuel is burnt.

OCR MEI M4 2010 June Q2

2 A particle of mass $m \mathrm {~kg}$ moves horizontally in a straight line with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t \mathrm {~s}$. The total resistance force on the particle is of magnitude $m k v ^ { \frac { 3 } { 2 } } \mathrm {~N}$ where $k$ is a positive constant. There are no other horizontal forces present. Initially $v = 25$ and the particle is at a point O .

Show that $v = 4 \left( k t + \frac { 2 } { 5 } \right) ^ { - 2 }$.
Find the displacement from O of the particle at time $t$.
Describe the motion of the particle as $t$ increases. Section B (48 marks)

OCR MEI M4 2010 June Q3

3 A uniform rod AB of mass $m$ and length $4 a$ is hinged at a fixed point C , where $\mathrm { AC } = a$, and can rotate freely in a vertical plane. A light elastic string of natural length $2 a$ and modulus $\lambda$ is attached at one end to B and at the other end 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 $2 a$ above C . The string is always vertical. This system is shown in Fig. 3 with the rod inclined at $\theta$ to the horizontal. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cb86219c-e0b1-4f75-b8b2-50b5a233aa54-2_387_613_1763_767} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Find an expression for $V$, the potential energy of the system relative to C , and show that $\frac { \mathrm { d } V } { \mathrm {~d} \theta } = a \cos \theta \left( \frac { 9 } { 2 } \lambda \sin \theta - m g \right)$.
Determine the positions of equilibrium and the nature of their stability in the cases
(A) $\lambda > \frac { 2 } { 9 } m g$,
(B) $\lambda < \frac { 2 } { 9 } m g$,
(C) $\lambda = \frac { 2 } { 9 } m g$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cb86219c-e0b1-4f75-b8b2-50b5a233aa54-3_522_755_342_696} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure}
Show, by integration, that the moment of inertia of the cone about its axis of symmetry is $\frac { 3 } { 10 } M a ^ { 2 }$. [You may assume the standard formula for the moment of inertia of a uniform circular disc about its axis of symmetry and the formula $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$ for the volume of a cone.] A frustum is made by taking a uniform cone of base radius 0.1 m and height 0.2 m and removing a cone of height 0.1 m and base radius 0.05 m as shown in Fig. 4.2. The mass of the frustum is 2.8 kg . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cb86219c-e0b1-4f75-b8b2-50b5a233aa54-3_391_517_1352_813} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
\end{figure} The frustum can rotate freely about its axis of symmetry which is fixed and vertical.
Show that the moment of inertia of the frustum about its axis of symmetry is $0.0093 \mathrm {~kg} \mathrm {~m} ^ { 2 }$. The frustum is accelerated from rest for $t$ seconds by a couple of magnitude 0.05 N m about its axis of symmetry, until it is rotating at $10 \mathrm { rad } \mathrm { s } ^ { - 1 }$.
Calculate $t$.
Find the position of G , the centre of mass of the frustum. The frustum (rotating at $10 \mathrm { rad } \mathrm { s } ^ { - 1 }$ ) then receives an impulse tangential to the circumference of the larger circular face. This reduces its angular speed to $5 \mathrm { rad } \mathrm { s } ^ { - 1 }$.
To reduce its angular speed further, a parallel impulse of the same magnitude is now applied tangentially in the horizontal plane through G at the curved surface of the frustum. Calculate the resulting angular speed.

OCR MEI M4 2011 June Q1

1 A raindrop of mass $m$ falls vertically from rest under gravity. Initially the mass of the raindrop is $m _ { 0 }$. As it falls it loses mass by evaporation at a rate $\lambda m$, where $\lambda$ is a constant. Its motion is modelled by assuming that the evaporation produces no resultant force on the raindrop. The velocity of the raindrop is $v$ at time $t$. The forces on the raindrop are its weight and a resistance force of magnitude $k m v$, where $k$ is a constant.

Find $m$ in terms of $m _ { 0 } , \lambda$ and $t$.
Write down the equation of motion of the raindrop. Solve this differential equation and hence show that $v = \frac { g } { \lambda - k } \left( \mathrm { e } ^ { ( \lambda - k ) t } - 1 \right)$.
Find the velocity of the raindrop when it has lost half of its initial mass.

OCR MEI M4 2011 June Q2

2 A small ring of mass $m$ can slide freely along a fixed smooth horizontal rod. A light elastic string of natural length $a$ and stiffness $k$ has one end attached to a point A on the rod and the other end attached to the ring. An identical elastic string has one end attached to the ring and the other end attached to a point B which is a distance $a$ vertically above the rod and a horizontal distance $2 a$ from the point A . The displacement of the ring from the vertical line through B is $x$, as shown in Fig. 2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0166dd50-5069-47f4-a015-d01a9c54faf4-2_405_1063_1270_539} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

Find an expression for $V$, the potential energy of the system when $0 < x < a$, and show that $$\frac { \mathrm { d } V } { \mathrm {~d} x } = 2 k x - k a - \frac { k a x } { \sqrt { a ^ { 2 } + x ^ { 2 } } }$$
Show that $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }$ is always positive.
Show that there is a position of equilibrium with $\frac { 1 } { 2 } a < x < a$. State, with a reason, whether it is stable or unstable.

OCR MEI M4 2011 June Q3

3 A car of mass 800 kg moves horizontally in a straight line with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t$ seconds. While $v \leqslant 20$, the power developed by the engine is $8 v ^ { 4 } \mathrm {~W}$. The total resistance force on the car is of magnitude $8 v ^ { 2 } \mathrm {~N}$. Initially $v = 2$ and the car is at a point O . At time $t$ s the displacement from O is $x \mathrm {~m}$.

Find $v$ in terms of $x$ and show that when $v = 20 , x = 100 \ln 1.9$.
Find the relationship between $t$ and $x$, and show that when $v = 20 , t \approx 19.2$. The driving force is removed at the instant when $v$ reaches 20 .
For the subsequent motion, find $v$ in terms of $t$. Calculate $t$ when $v = 2$.

OCR MEI M4 2011 June Q4

4 In this question you may assume without proof the standard results in Examination Formulae and Tables (MF2) for

the moment of inertia of a disc about an axis through its centre perpendicular to the disc,
the position of the centre of mass of a solid uniform cone.

Fig. 4 shows a uniform cone of radius $a$ and height $2 a$, with its axis of symmetry on the $x$-axis and its vertex at the origin. A thin slice through the cone parallel to the base is at a distance $x$ from the vertex. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0166dd50-5069-47f4-a015-d01a9c54faf4-3_497_748_1283_699} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} The slice is taken to be a thin uniform disc of mass $m$.

Write down the moment of inertia of the disc about the $x$-axis. Hence show that the moment of inertia of the disc about the $y$-axis is $\frac { 17 } { 16 } m x ^ { 2 }$.
Hence show by integration that the moment of inertia of the cone about the $y$-axis is $\frac { 51 } { 20 } M a ^ { 2 }$, where $M$ is the mass of the cone. [You may assume without proof the formula for the volume of a cone.] The cone is now suspended so that it can rotate freely about a fixed, horizontal axis through its vertex. The axis of symmetry of the cone moves in a vertical plane perpendicular to the axis of rotation. The cone is released from rest when its axis of symmetry is at an acute angle $\alpha$ to the downward vertical. At time $t$, the angle the axis of symmetry makes with the downward vertical is $\theta$.
Use an energy method to show that $\dot { \theta } ^ { 2 } = \frac { 20 g } { 17 a } ( \cos \theta - \cos \alpha )$.
Hence, or otherwise, show that if $\alpha$ is small the cone performs approximate simple harmonic motion and find the period. RECOGNISING ACHIEVEMENT

OCR MEI M4 2012 June Q1

1 A rocket in deep space has initial mass $m _ { 0 }$ and is moving in a straight line at speed $v _ { 0 }$. It fires its engine in the direction opposite to the motion in order to increase its speed. The propulsion system ejects matter at a constant mass rate $k$ with constant speed $u$ relative to the rocket. At time $t$ after the engines are fired, the speed of the rocket is $v$.

Show that while mass is being ejected from the rocket, $\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k$.
Hence find an expression for $v$ at time $t$.

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OCR MEI M4 2007 June Q4

OCR MEI M4 2008 June Q1

OCR MEI M4 2008 June Q2

OCR MEI M4 2008 June Q3

OCR MEI M4 2008 June Q4

OCR MEI M4 2009 June Q1

OCR MEI M4 2009 June Q2

OCR MEI M4 2009 June Q3

OCR MEI M4 2009 June Q4

OCR MEI M4 2010 June Q1

OCR MEI M4 2010 June Q2

OCR MEI M4 2010 June Q3

OCR MEI M4 2011 June Q1

OCR MEI M4 2011 June Q2

OCR MEI M4 2011 June Q3

OCR MEI M4 2011 June Q4

OCR MEI M4 2012 June Q1