Questions - OCR MEI M3 - A-Level Maths

OCR MEI M3 2010 June Q4

4 A particle P is performing simple harmonic motion in a vertical line. At time $t \mathrm {~s}$, its displacement $x \mathrm {~m}$ above a fixed point O is given by $$x = A \sin \omega t + B \cos \omega t$$ where $A , B$ and $\omega$ are constants.

Show that the acceleration of P , in $\mathrm { ms } ^ { - 2 }$, is $- \omega ^ { 2 } x$. When $t = 0 , \mathrm { P }$ is 16 m below O , moving with velocity $7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ upwards, and has acceleration $1 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ upwards.
Find the values of $A , B$ and $\omega$.
Find the maximum displacement, the maximum speed, and the maximum acceleration of P .
Find the speed and the direction of motion of P when $t = 15$.
Find the distance travelled by P between $t = 0$ and $t = 15$.

OCR MEI M3 2011 June Q2

2

A particle P of mass 0.2 kg is connected to a fixed point O by a light inextensible string of length 3.2 m , and is moving in a vertical circle with centre O and radius 3.2 m . Air resistance may be neglected. When P is at the highest point of the circle, the tension in the string is 0.6 N .
1. Find the speed of P when it is at the highest point.
2. For an instant when OP makes an angle of $60 ^ { \circ }$ with the downward vertical, find
  (A) the radial and tangential components of the acceleration of P ,
  (B) the tension in the string.
A solid cone is fixed with its axis of symmetry vertical and its vertex V uppermost. The semivertical angle of the cone is $36 ^ { \circ }$, and its surface is smooth. A particle Q of mass 0.2 kg is connected to V by a light inextensible string, and Q moves in a horizontal circle at constant speed, in contact with the surface of the cone, as shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-3_455_609_950_808} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The particle Q makes one complete revolution in 1.8 s , and the normal reaction of the cone on Q has magnitude 0.75 N .
1. Find the tension in the string.
2. Find the length of the string.

OCR MEI M3 2011 June Q3

3 Fixed points A and B are 4.8 m apart on the same horizontal level. The midpoint of AB is M . A light elastic string, with natural length 3.9 m and modulus of elasticity 573.3 N , has one end attached to A and the other end attached to $\mathbf { B }$.

Find the elastic energy stored in the string. A particle P is attached to the midpoint of the string, and is released from rest at M . It comes instantaneously to rest when P is 1.8 m vertically below M .
Show that the mass of P is 15 kg .
Verify that P can rest in equilibrium when it is 1.0 m vertically below M . In general, a light elastic string, with natural length $a$ and modulus of elasticity $\lambda$, has its ends attached to fixed points which are a distance $d$ apart on the same horizontal level. A particle of mass $m$ is attached to the midpoint of the string, and in the equilibrium position each half of the string has length $h$, as shown in Fig. 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-4_280_755_1064_696} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} When the particle makes small oscillations in a vertical line, the period of oscillation is given by the formula $$\sqrt { \frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } } } m ^ { \alpha } a ^ { \beta } \lambda ^ { \gamma }$$
Show that $\frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } }$ is dimensionless.
Use dimensional analysis to find $\alpha , \beta$ and $\gamma$.
Hence find the period when the particle P makes small oscillations in a vertical line centred on the position of equilibrium given in part (iii).

OCR MEI M3 2011 June Q4

4 The region $A$ is bounded by the curve $y = x ^ { 2 } + 5$ for $0 \leqslant x \leqslant 3$, the $x$-axis, the $y$-axis and the line $x = 3$. The region $B$ is bounded by the curve $y = x ^ { 2 } + 5$ for $0 \leqslant x \leqslant 3$, the $y$-axis and the line $y = 14$. These regions are shown in Fig. 4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-5_883_554_431_794} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Find the coordinates of the centre of mass of a uniform lamina occupying the region $A$.
The region $B$ is rotated through $2 \pi$ radians about the $y$-axis to form a uniform solid of revolution $R$. Find the $y$-coordinate of the centre of mass of the solid $R$.
The region $A$ is rotated through $2 \pi$ radians about the $y$-axis to form a uniform solid of revolution $S$. Using your answer to part (ii), or otherwise, find the $y$-coordinate of the centre of mass of the solid $S$.

OCR MEI M3 2012 June Q1

1 The fixed point A is at a height $4 b$ above a smooth horizontal surface, and C is the point on the surface which is vertically below A. A light elastic string, of natural length $3 b$ and modulus of elasticity $\lambda$, has one end attached to A and the other end attached to a block of mass $m$. The block is released from rest at a point B on the surface where $\mathrm { BC } = 3 b$, as shown in Fig. 1. You are given that the block remains on the surface and moves along the line BC . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-2_511_887_488_589} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure}

Show that immediately after release the acceleration of the block is $\frac { 2 \lambda } { 5 m }$.
Show that, when the block reaches C , its speed $v$ is given by $v ^ { 2 } = \frac { \lambda b } { m }$.
Show that the equation $v ^ { 2 } = \frac { \lambda b } { m }$ is dimensionally consistent. The time taken for the block to move from B to C is given by $k m ^ { \alpha } b ^ { \beta } \lambda ^ { \gamma }$, where $k$ is a dimensionless constant.
Use dimensional analysis to find $\alpha , \beta$ and $\gamma$. When the string has natural length 1.2 m and modulus of elasticity 125 N , and the block has mass 8 kg , the time taken for the block to move from B to C is 0.718 s .
Find the time taken for the block to move from B to C when the string has natural length 9 m and modulus of elasticity 20 N , and the block has mass 75 kg .

OCR MEI M3 2012 June Q2

2

Fig. 2 shows a car of mass 800 kg moving at constant speed in a horizontal circle with centre C and radius 45 m , on a road which is banked at an angle of $18 ^ { \circ }$ to the horizontal. The forces shown are the weight $W$ of the car, the normal reaction, $R$, of the road on the car and the frictional force $F$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-3_286_970_402_561} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
1. Given that the frictional force is zero, find the speed of the car.
2. Given instead that the speed of the car is $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find the frictional force and the normal reaction.
One end of a light inextensible string is attached to a fixed point O , and the other end is attached to a particle P of mass $m \mathrm {~kg}$. Starting with the string taut and P vertically below $\mathrm { O } , \mathrm { P }$ is set in motion with a horizontal velocity of $7 \mathrm {~ms} ^ { - 1 }$. It then moves in part of a vertical circle with centre O . The string becomes slack when the speed of P is $2.8 \mathrm {~ms} ^ { - 1 }$. Find the length of the string. Find also the angle that OP makes with the upward vertical at the instant when the string becomes slack.

OCR MEI M3 2012 June Q3

3 A particle Q is performing simple harmonic motion in a vertical line. Its height, $x$ metres, above a fixed level at time $t$ seconds is given by $$x = c + A \cos ( \omega t - \phi )$$ where $c , A , \omega$ and $\phi$ are constants.

Show that $\ddot { x } = - \omega ^ { 2 } ( x - c )$. Fig. 3 shows the displacement-time graph of Q for $0 \leqslant t \leqslant 14$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-4_547_1079_703_495} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
Find exact values for $c , A , \omega$ and $\phi$.
Find the maximum speed of Q .
Find the height and the velocity of Q when $t = 0$.
Find the distance travelled by Q between $t = 0$ and $t = 14$.

OCR MEI M3 2012 June Q4

4

A uniform lamina occupies the region bounded by the $x$-axis, the $y$-axis and the curve $y = 3 - \sqrt { x }$ for $0 \leqslant x \leqslant 9$. Find the coordinates of the centre of mass of this lamina.
Fig. 4.1 shows the region bounded by the line $x = 2$ and the part of the circle $y ^ { 2 } = 25 - x ^ { 2 }$ for which $2 \leqslant x \leqslant 5$. This region is rotated about the $x$-axis to form a uniform solid of revolution $S$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_675_659_479_705} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure}
1. Find the $x$-coordinate of the centre of mass of $S$. The solid $S$ rests in equilibrium with its curved surface in contact with a rough plane inclined at $25 ^ { \circ }$ to the horizontal. Fig. 4.2 shows a vertical section containing AB , which is a diameter and also a line of greatest slope of the flat surface of $S$. This section also contains XY, which is a line of greatest slope of the plane. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_494_560_1615_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
  \end{figure}
2. Find the angle $\theta$ that AB makes with the horizontal.

OCR MEI M3 2013 June Q1

1

A particle P of mass 1.5 kg is connected to a fixed point by a light inextensible string of length 3.2 m . The particle P is moving as a conical pendulum in a horizontal circle at a constant angular speed of $2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }$.
1. Find the tension in the string.
2. Find the angle that the string makes with the vertical.
A particle Q of mass $m$ moves on a smooth horizontal surface, and is connected to a fixed point on the surface by a light elastic string of natural length $d$ and stiffness $k$. With the string at its natural length, Q is set in motion with initial speed $u$ perpendicular to the string. In the subsequent motion, the maximum length of the string is $2 d$, and the string first returns to its natural length after time $t$. You are given that $u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }$ and $t = A k ^ { \alpha } d ^ { \beta } m ^ { \gamma }$, where $A$ is a dimensionless constant.
1. Show that the dimensions of $k$ are $\mathrm { MT } ^ { - 2 }$.
2. Show that the equation $u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }$ is dimensionally consistent.
3. Find $\alpha , \beta$ and $\gamma$. You are now given that Q has mass 5 kg , and the string has natural length 0.7 m and stiffness $60 \mathrm { Nm } ^ { - 1 }$.
4. Find the initial speed $u$, and use conservation of energy to find the speed of Q at the instant when the length of the string is double its natural length.

OCR MEI M3 2013 June Q2

2 A particle P of mass 0.25 kg is connected to a fixed point O by a light inextensible string of length $a$ metres, and is moving in a vertical circle with centre O and radius $a$ metres. When P is vertically below O , its speed is $8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When OP makes an angle $\theta$ with the downward vertical, and the string is still taut, P has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the tension in the string is $T \mathrm {~N}$, as shown in Fig. 2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-3_483_551_447_749} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

Find an expression for $v ^ { 2 }$ in terms of $a$ and $\theta$, and show that $$T = \frac { 17.64 } { a } + 7.35 \cos \theta - 4.9 .$$
Given that $a = 0.9$, show that P moves in a complete circle. Find the maximum and minimum magnitudes of the tension in the string.
Find the largest value of $a$ for which P moves in a complete circle.
Given that $a = 1.6$, find the speed of P at the instant when the string first becomes slack.

OCR MEI M3 2013 June Q3

3 A light spring, with modulus of elasticity 686 N , has one end attached to a fixed point A . The other end is attached to a particle P of mass 18 kg which hangs in equilibrium when it is 2.2 m vertically below A .

Find the natural length of the spring AP . Another light spring has natural length 2.5 m and modulus of elasticity 145 N . One end of this spring is now attached to the particle P , and the other end is attached to a fixed point B which is 2.5 m vertically below P (so leaving the equilibrium position of P unchanged). While in its equilibrium position, P is set in motion with initial velocity $3.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ vertically downwards, as shown in Fig. 3. It now executes simple harmonic motion along part of the vertical line AB . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-4_721_383_726_831} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} At time $t$ seconds after it is set in motion, P is $x$ metres below its equilibrium position.
Show that the tension in the spring AP is $( 176.4 + 392 x ) \mathrm { N }$, and write down an expression for the thrust in the spring BP.
Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 25 x$.
Find the period and the amplitude of the motion.
Find the magnitude and direction of the velocity of P when $t = 2.4$.
Find the total distance travelled by P during the first 2.4 seconds of its motion.

OCR MEI M3 2013 June Q4

4

A uniform solid of revolution $S$ is formed by rotating the region enclosed between the $x$-axis and the curve $y = x \sqrt { 4 - x }$ for $0 \leqslant x \leqslant 4$ through $2 \pi$ radians about the $x$-axis, as shown in Fig. 4.1. O is the origin and the end A of the solid is at the point $( 4,0 )$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_520_625_408_703} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure}
1. Find the $x$-coordinate of the centre of mass of the solid $S$. The solid $S$ has weight $W$, and it is freely hinged to a fixed point at O . A horizontal force, of magnitude $W$ acting in the vertical plane containing OA , is applied at the point A , as shown in Fig. 4.2. $S$ is in equilibrium. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_346_512_1361_781} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
  \end{figure}
2. Find the angle that OA makes with the vertical.
  [0pt] [Question 4(b) is printed overleaf]
Fig. 4.3 shows the region bounded by the $x$-axis, the $y$-axis, the line $y = 8$ and the curve $y = ( x - 2 ) ^ { 3 }$ for $0 \leqslant y \leqslant 8$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-6_631_695_370_683} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
\end{figure} Find the coordinates of the centre of mass of a uniform lamina occupying this region.

OCR MEI M3 2014 June Q1

1

The speed $v$ of sound in a solid material is given by $v = \sqrt { \frac { E } { \rho } }$, where $E$ is Young's modulus for the material and $\rho$ is its density.
1. Find the dimensions of Young's modulus. The density of steel is $7800 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$ and the speed of sound in steel is $6100 \mathrm {~ms} ^ { - 1 }$.
2. Find Young's modulus for steel, stating the units in which your answer is measured. A tuning fork has cylindrical prongs of radius $r$ and length $l$. The frequency $f$ at which the tuning fork vibrates is given by $f = k c ^ { \alpha } E ^ { \beta } \rho ^ { \gamma }$, where $c = \frac { l ^ { 2 } } { r }$ and $k$ is a dimensionless constant.
3. Find $\alpha , \beta$ and $\gamma$.
A particle P is performing simple harmonic motion along a straight line, and the centre of the oscillations is O . The points X and Y on the line are on the same side of O , at distances 3.9 m and 6.0 m from O respectively. The speed of P is $1.04 \mathrm {~ms} ^ { - 1 }$ when it passes through X and $0.5 \mathrm {~ms} ^ { - 1 }$ when it passes through Y.
1. Find the amplitude and the period of the oscillations.
2. Find the time taken for P to travel directly from X to Y .

OCR MEI M3 2014 June Q2

2

The fixed point A is vertically above the fixed point B . A light inextensible string of length 5.4 m has one end attached to A and the other end attached to B. The string passes through a small smooth ring R of mass 0.24 kg , and R is moving at constant angular speed in a horizontal circle. The circle has radius 1.6 m , and $\mathrm { AR } = 3.4 \mathrm {~m} , \mathrm { RB } = 2.0 \mathrm {~m}$, as shown in Fig. 2 . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-3_565_504_447_753} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
1. Find the tension in the string.
2. Find the angular speed of R .
A particle P of mass 0.3 kg is joined to a fixed point O by a light inextensible string of length 1.8 m . The particle P moves without resistance in part of a vertical circle with centre O and radius 1.8 m . When OP makes an angle of $25 ^ { \circ }$ with the downward vertical, the tension in the string is 15 N .
1. Find the speed of P when OP makes an angle of $25 ^ { \circ }$ with the downward vertical.
2. Find the tension in the string when OP makes an angle of $60 ^ { \circ }$ with the upward vertical.
3. Find the speed of P at the instant when the string becomes slack.

OCR MEI M3 2014 June Q3

3 The fixed points A and B lie on a line of greatest slope of a smooth inclined plane, with B higher than A . The horizontal distance from A to B is 2.4 m and the vertical distance is 0.7 m . The fixed point C is 2.5 m vertically above B . A light elastic string of natural length 2.2 m has one end attached to C and the other end attached to a small block of mass 9 kg which is in contact with the plane. The block is in equilibrium when it is at A, as shown in Fig. 3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-4_712_641_488_687} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Show that the modulus of elasticity of the string is 37.73 N . The block starts at A and is at rest. A constant force of 18 N , acting in the direction AB , is then applied to the block so that it slides along the line AB .
Find the magnitude and direction of the acceleration of the block
(A) when it leaves the point A ,
(B) when it reaches the point B .
Find the speed of the block when it reaches the point B .

OCR MEI M3 2014 June Q4

4 The region $R$ is bounded by the $x$-axis, the $y$-axis, the curve $y = \mathrm { e } ^ { - x }$ and the line $x = k$, where $k$ is a positive constant.

The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a uniform solid of revolution. Find the $x$-coordinate of the centre of mass of this solid, and show that it can be written in the form $$\frac { 1 } { 2 } - \frac { k } { \mathrm { e } ^ { 2 k } - 1 } .$$
The solid in part (i) is placed with its larger circular face in contact with a rough plane inclined at $60 ^ { \circ }$ to the horizontal, as shown in Fig. 4, and you are given that no slipping occurs. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-5_508_483_712_790} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Show that, whatever the value of $k$, the solid will not topple.
A uniform lamina occupies the region $R$. Find, in terms of $k$, the coordinates of the centre of mass of this lamina. \section*{END OF QUESTION PAPER}

OCR MEI M3 2015 June Q2

2

A particle P of mass $m$ is attached to a fixed point O by a light inextensible string of length $a$. P is moving without resistance in a complete vertical circle with centre O and radius $a$. When P is at the highest point of the circle, the tension in the string is $T _ { 1 }$. When OP makes an angle $\theta$ with the upward vertical, the tension in the string is $T _ { 2 }$. Show that $$T _ { 2 } = T _ { 1 } + 3 m g ( 1 - \cos \theta ) .$$
The fixed point A is 1.2 m vertically above the fixed point C . A particle Q of mass 0.9 kg is joined to A , to C , and to a particle R of mass 1.5 kg , by three light inextensible strings of lengths $1.3 \mathrm {~m} , 0.5 \mathrm {~m}$ and 1.8 m respectively. The particle Q moves in a horizontal circle with centre C , and R moves in a horizontal circle at the same constant angular speed as Q , in such a way that $\mathrm { A } , \mathrm { C } , \mathrm { Q }$ and R are always coplanar. The string QR makes an angle of $60 ^ { \circ }$ with the downward vertical. This situation is shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-3_579_1191_881_406} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
1. Find the tensions in the strings QR and AQ .
2. Find the angular speed of the system.
3. Find the tension in the string CQ .

OCR MEI M3 2015 June Q3

3 Fig. 3 shows the fixed points A and F which are 9.5 m apart on a smooth horizontal surface and points B and D on the line AF such that $\mathrm { AB } = \mathrm { DF } = 3.0 \mathrm {~m}$. A small block of mass 10.5 kg is joined to A by a light elastic string of natural length 3.0 m and stiffness $12 \mathrm { Nm } ^ { - 1 }$; the block is joined to F by a light elastic string of natural length 3.0 m and stiffness $30 \mathrm { Nm } ^ { - 1 }$. The block is released from rest at B and then slides along part of the line AF . The block has zero acceleration when it is at a point C , and it comes to instantaneous rest at a point E . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-4_221_1082_536_502} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Find the distance BC . At time $t \mathrm {~s}$ the displacement of the block from C is $x \mathrm {~m}$, measured in the direction AF .
Show that, when the block is between B and $\mathrm { D } , \frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 4 x$.
Find the maximum speed of the block.
Find the distance of the block from C when its speed is $4.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the time taken for the block to travel from B to D.
Find the distance DE .

OCR MEI M3 2015 June Q4

4

A uniform lamina occupies the region bounded by the $x$-axis and the curve $y = \frac { x ^ { 2 } ( a - x ) } { a ^ { 2 } }$ for $0 \leqslant x \leqslant a$. Find the coordinates of the centre of mass of this lamina.
The region $A$ is bounded by the $x$-axis, the $y$-axis, the curve $y = \sqrt { x ^ { 2 } + 16 }$ and the line $x = 3$. The region $B$ is bounded by the $y$-axis, the curve $y = \sqrt { x ^ { 2 } + 16 }$ and the line $y = 5$. These regions are shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-5_604_460_605_792} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
1. Find the $x$-coordinate of the centre of mass of the uniform solid of revolution formed when the region $A$ is rotated through $2 \pi$ radians about the $x$-axis.
2. Using your answer to part (i), or otherwise, find the $x$-coordinate of the centre of mass of the uniform solid of revolution formed when the region $B$ is rotated through $2 \pi$ radians about the $x$-axis. \section*{END OF QUESTION PAPER}

OCR MEI M3 2016 June Q1

1

In an investigation, small spheres are dropped into a long column of a viscous liquid and their terminal speeds measured. It is thought that the terminal speed $V$ of a sphere depends on a product of powers of its radius $r$, its weight $m g$ and the viscosity $\eta$ of the liquid, and is given by $$V = k r ^ { \alpha } ( m g ) ^ { \beta } \eta ^ { \gamma } ,$$ where $k$ is a dimensionless constant.
1. Given that the dimensions of viscosity are $\mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 1 }$ find $\alpha , \beta$ and $\gamma$. A sphere of mass 0.03 grams and radius 0.2 cm has a terminal speed of $6 \mathrm {~ms} ^ { - 1 }$ when falling through a liquid with viscosity $\eta$. A second sphere of radius 0.25 cm falling through the same liquid has a terminal speed of $8 \mathrm {~ms} ^ { - 1 }$.
2. Find the mass of the second sphere.
A manufacturer is testing different types of light elastic ropes to be used in bungee jumping. You may assume that air resistance is negligible. A bungee jumper of mass 80 kg is connected to a fixed point A by one of these elastic ropes. The natural length of this rope is 25 m and its modulus of elasticity is 1600 N . At one instant, the jumper is 30 m directly below A and he is moving vertically upwards at $15 \mathrm {~ms} ^ { - 1 }$. He comes to instantaneous rest at a point B , with the rope slack.
1. Find the distance AB . The same bungee jumper now tests a second rope, also of natural length 25 m . He falls from rest at A . It is found that he first comes instantaneously to rest at a distance 54 m directly below A .
2. Find the modulus of elasticity of this second rope. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-3_559_705_262_680} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
  \end{figure} The region R shown in Fig. 2.1 is bounded by the curve $y = k ^ { 2 } - x ^ { 2 }$, for $0 \leqslant x \leqslant k$, and the coordinate axes. The $x$-coordinate of the centre of mass of a uniform lamina occupying the region R is 0.75 .
3. Show that $k = 2$. A uniform solid S is formed by rotating the region R through $2 \pi$ radians about the $x$-axis.
4. Show that the centre of mass of S is at $( 0.625,0 )$. Fig. 2.2 shows a solid T made by attaching the solid S to the base of a uniform solid circular cone C . The cone $C$ is made of the same material as $S$ and has height 8 cm and base radius 4 cm . \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-3_455_794_1521_639} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
  \end{figure}
5. Show that the centre of mass of T is at a distance of 6.75 cm from the vertex of the cone. [You may quote the standard results that the volume of a cone is $\frac { 1 } { 3 } \pi r ^ { 2 } h$ and its centre of mass is $\frac { 3 } { 4 } h$ from its vertex.]
6. The solid T is suspended from a point P on the circumference of the base of C . Find the acute angle between the axis of symmetry of T and the vertical. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-4_668_262_255_904} \captionsetup{labelformat=empty} \caption{Fig. 3}
  \end{figure} One end of a light elastic string, of natural length 2.7 m and modulus of elasticity 54 N , is attached to a fixed point L . The other end of the string is attached to a particle P of mass 2.5 kg . One end of a second light elastic string, of natural length 1.7 m and modulus of elasticity 8.5 N , is attached to P . The other end of this second string is attached to a fixed point M , which is 6 m vertically below L . This situation is shown in Fig. 3. The particle P is released from rest when it is 4.2 m below L . Both strings remain taut throughout the subsequent motion. At time $t \mathrm {~s}$ after P is released from rest, its displacement below L is $x \mathrm {~m}$.
7. Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 10 ( x - 4 )$.
8. Write down the value of $x$ when P is at the centre of its motion.
9. Find the amplitude and the period of the oscillations.
10. Find the velocity of P when $t = 1.2$.

OCR MEI M3 2016 June Q4

4 A particle P of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point O . Particle P is projected so that it moves in complete vertical circles with centre O ; there is no air resistance. A and B are two points on the circle, situated on opposite sides of the vertical through O . The lines OA and OB make angles $\alpha$ and $\beta$ with the upward vertical as shown in Fig. 4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-5_414_399_434_833} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} The speed of P at A is $\sqrt { \frac { 17 a g } { 3 } }$. The speed of P at B is $\sqrt { 5 a g }$ and $\cos \beta = \frac { 2 } { 3 }$.

Show that $\cos \alpha = \frac { 1 } { 3 }$. On one occasion, when P is at its lowest point and moving in a clockwise direction, it collides with a stationary particle Q . The two particles coalesce and the combined particle continues to move in the same vertical circle. When this combined particle reaches the point A , the string becomes slack.
Show that when the string becomes slack, the speed of the combined particle is $\sqrt { \frac { a g } { 3 } }$. The mass of the particle Q is $k m$.
Find the value of $k$.
Find, in terms of $m$ and $g$, the instantaneous change in the tension in the string as a result of the collision.

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