Questions M3 - A-Level Maths

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.

Edexcel M3 Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-2_341_652_217_621} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A particle of mass 0.6 kg is attached to one end of a light elastic spring of natural length 1 m and modulus of elasticity 30 N . The other end of the spring is fixed to a point $O$ which lies on a smooth plane inclined at an angle $\alpha$ to the horizontal where $\tan \alpha = \frac { 3 } { 4 }$ as shown in Figure 1. The particle is held at rest on the slope at a point 1.2 m from $O$ down the line of greatest slope of the plane.

Find the tension in the spring.
Find the initial acceleration of the particle.

Edexcel M3 Q2

2. A particle $P$ of mass 0.5 kg moves along the positive $x$-axis under the action of a single force directed away from the origin $O$. When $P$ is $x$ metres from $O$, the magnitude of the force is $3 x ^ { \frac { 1 } { 2 } } \mathrm {~N}$ and $P$ has a speed of $v \mathrm {~ms} ^ { - 1 }$. Given that when $x = 1 , P$ is moving away from $O$ with speed $2 \mathrm {~ms} ^ { - 1 }$,

find an expression for $v ^ { 2 }$ in terms of $x$,
show that when $x = 4 , P$ has a speed of $7.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 1 decimal place.

Edexcel M3 Q3

3. A particle is performing simple harmonic motion along a straight line between the points $A$ and $B$ where $A B = 8 \mathrm {~m}$. The period of the motion is 12 seconds.

Find the maximum speed of the particle in terms of $\pi$. The points $P$ and $Q$ are on the line $A B$ at distances of 3 m and 6 m respectively from $A$.
Find, correct to 3 significant figures, the time it takes for the particle to travel directly from $P$ to $Q$.
(6 marks)

Edexcel M3 Q4

4. Whilst in free-fall a parachutist falls vertically such that his velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when he is $x$ metres below his initial position is given by $$v ^ { 2 } = k g \left( 1 - \mathrm { e } ^ { - \frac { 2 x } { k } } \right) ,$$ where $k$ is a constant.
Given that he experiences an acceleration of $\mathrm { f } \mathrm { m } \mathrm { s } ^ { - 2 }$,

show that $f = g \mathrm { e } ^ { - \frac { 2 x } { k } }$. After falling a large distance, his velocity is constant at $49 \mathrm {~ms} ^ { - 1 }$.
Find the value of $k$.
Hence, express $f$ in the form ( $\lambda - \mu v ^ { 2 }$ ) where $\lambda$ and $\mu$ are constants which you should find.
(4 marks)

Edexcel M3 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-3_588_291_1126_662} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A firework is modelled as a uniform solid formed by joining the plane surface of a right circular cone of height $2 r$ and base radius $r$, to one of the plane surfaces of a cylinder of height $h$ and base radius $r$ as shown in Figure 2. Using this model,

show that the distance of the centre of mass of the firework from its plane base is $$\frac { 3 h ^ { 2 } + 4 h r + 2 r ^ { 2 } } { 2 ( 3 h + 2 r ) }$$ The firework is to be launched from rough ground inclined at an angle $\alpha$ to the horizontal. Given that the firework does not slip or topple and that $h = 4 r$,
Find, correct to the nearest degree, the maximum value of $\alpha$.

Edexcel M3 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-4_437_364_196_717} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} The two ends of a light inextensible string of length $3 a$ are attached to fixed points $Q$ and $R$ which are a distance of $a \sqrt { } 3$ apart with $R$ vertically below $Q$. A particle $P$ of mass $m$ is attached to the string at a distance of $2 a$ from $Q$.
$P$ is given a horizontal speed, $u$, such that it moves in a horizontal circle with both sections of the string taut as shown in Figure 3.

Show that $\angle P R Q$ is a right angle.
Find $\angle P Q R$ in degrees.
Find, in terms of $a , g , m$ and $u$, the tension in the section of string
1. $P Q$,
2. $P R$.
Show that $u ^ { 2 } \geq \frac { g a } { \sqrt { 3 } }$.

Edexcel M3 Q7

7. A particle of mass 2 kg is attached to one end of a light elastic string of natural length 1 m and modulus of elasticity 50 N . The other end of the string is attached to a fixed point $O$ on a rough horizontal plane and the coefficient of friction between the particle and the plane is $\frac { 10 } { 49 }$. The particle is projected from $O$ along the plane with an initial speed of $5 \mathrm {~ms} ^ { - 1 }$.

Show that the greatest distance from $O$ which the particle reaches is 1.84 m .
Find, correct to 2 significant figures, the speed at which the particle returns to $O$.

Edexcel M3 Q1

A student is attempting to model the expansion of an airbag in a car following a collision.

The student considers the displacement from the steering column, $s$ metres, of a point $P$ on the airbag $t$ seconds after a collision and uses the formula $$s = \mathrm { e } ^ { 3 t } - 1 , \quad 0 \leq t \leq 0.1$$ Using this model,

find, correct to the nearest centimetre, the maximum displacement of $P$,
find the initial velocity of $P$,
find the acceleration of $P$ in terms of $t$.
Explain why this model is unlikely to be realistic.

Edexcel M3 Q2

2. A particle $P$ is attached to one end of a light elastic string of modulus of elasticity 80 N . The other end of the string is attached to a fixed point $A$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-2_410_570_1210_735} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} When a horizontal force of magnitude 20 N is applied to $P$, it rests in equilibrium with the string making an angle of $30 ^ { \circ }$ with the vertical and $A P = 1.2 \mathrm {~m}$ as shown in Figure 1.

Find the tension in the string.
Find the elastic potential energy stored in the string.

Edexcel M3 Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-3_513_570_196_625} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A particle of mass $m$ is suspended at a point $A$ vertically below a fixed point $O$ by a light inextensible string of length $a$ as shown in Figure 2. The particle is given a horizontal velocity $u$ and subsequently moves along a circular arc until it reaches the point $B$ where the string becomes slack. Given that the point $B$ is at a height $\frac { 1 } { 2 } a$ above the level of $O$,

show that $\angle B O A = 120 ^ { \circ }$,
show that $u ^ { 2 } = \frac { 7 } { 2 } g a$.

Edexcel M3 Q4

4. On a particular day, high tide at the entrance to a harbour occurs at 11 a.m. and the water depth is 14 m . Low tide occurs $6 \frac { 1 } { 4 }$ hours later at which time the water depth is 6 m . In a model of the situation, the water level is assumed to perform simple harmonic motion.
Using this model,

write down the amplitude and period of the motion. A ship needs a depth of 9 m before it can enter or leave the harbour.
Show that on this day a ship must enter the harbour by 2.38 p.m., correct to the nearest minute, or wait for low tide to pass.
(6 marks)
Given that a ship is not ready to enter the harbour until 5 p.m.,
find, to the nearest minute, how long the ship must wait before it can enter the harbour.

Questions M3 (745 questions)

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