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Edexcel M3 2021 January Q4

9 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_442_506_251_721} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform right solid cone $C$ has diameter $6 a$ and height $8 a$, as shown in Figure 3.
The solid $S$ is formed by removing a cone of height $4 a$ from the top of $C$ and then removing an identical, inverted cone. The vertex of the removed cone is at the point $O$ in the centre of the base of $C$, as shown in Figure 4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_236_502_1126_721} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure}

Find the distance of the centre of mass of $S$ from $O$.
(5) The point $A$ lies on the circumference of the base of $S$ and the point $B$ lies on the circumference of the top of $S$. The points $O$, $A$ and $B$ all lie in the same vertical plane, as shown in Figure 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_248_449_1845_749} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The solid $S$ is freely suspended from the point $B$ and hangs in equilibrium.
Find the size of the angle that $A B$ makes with the downward vertical.

Edexcel M3 2021 January Q5

13 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-16_720_232_251_858} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} The fixed points, $A$ and $B$, are a distance $10 a$ apart, with $B$ vertically above $A$. One end of a light elastic string, of natural length $2 a$ and modulus of elasticity $2 m g$, is attached to a particle $P$ of mass $m$ and the other end is attached to $A$. One end of another light elastic string, of natural length $4 a$ and modulus of elasticity $6 m g$, is attached to $P$ and the other end is attached to $B$. The particle $P$ rests in equilibrium at the point $C$, as shown in Figure 6.

Show that each string has an extension of $2 a$.
(5) The particle $P$ is now pulled down vertically, so that it is a distance $a$ below $C$ and then released from rest.
Show that in the subsequent motion, $P$ performs simple harmonic motion.
Find, in terms of $a$ and $g$, the speed of $P$ when it is a distance $\frac { 7 } { 2 } a$ above $A$.

Edexcel M3 2021 January Q6

13 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-20_789_858_121_536} \captionsetup{labelformat=empty} \caption{Figure 7}

\end{figure} A particle of mass $m$ is attached to one end of a light inextensible string of length $2 a$. The other end of the string is attached to a fixed point $O$. The particle is initially held at the point $A$ with the string taut and $O A$ making an angle of $60 ^ { \circ }$ with the downward vertical. The particle is then projected upwards with a speed of $3 \sqrt { a g }$, perpendicular to $O A$, in the vertical plane containing $O A$, as shown in Figure 7. In an initial model of the motion of the particle, it is assumed that the string does not break. Using this model,

show that the particle performs complete vertical circles. In a refined model it is assumed that the string will break if the tension in it exceeds 7 mg . Using this refined model,
show that the particle still performs complete vertical circles. \includegraphics[max width=\textwidth, alt={}, center]{8a687d17-ec7e-463f-84dd-605f5c230db1-20_2249_50_314_1982}

Edexcel M3 2021 January Q7

11 marks Challenging +1.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-24_394_1027_248_461} \captionsetup{labelformat=empty} \caption{Figure 8}

\end{figure} A particle $P$ of mass 0.5 kg is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity 3 N . The other end of the string is attached to a fixed point $O$ on a rough plane. The plane is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 2 } { 7 }$ The coefficient of friction between $P$ and the plane is $\frac { \sqrt { 5 } } { 5 }$ The particle $P$ is initially at rest at the point $O$, as shown in Figure 8. The particle $P$ then receives an impulse of magnitude 4 Ns, directed up a line of greatest slope of the plane. The particle $P$ moves up the plane and comes to rest at the point $A$.

Find the extension of the elastic string when $P$ is at $A$.
Show that the particle does not remain at rest at $A$.

Edexcel M3 2022 January Q1

6 marks Standard +0.3

A light elastic string $A B$ has natural length $11 a$ and modulus of elasticity $6 m g$

A particle of mass $4 m$ is attached to the point $C$ on the string where $A C = 8 a$ and a particle of mass $2 m$ is attached to the end $B$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-02_581_202_429_957} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The end $A$ of the string is attached to a fixed point and the string hangs vertically below $A$ with the particle of mass $4 m$ in equilibrium at the point $P$ and the particle of mass $2 m$ in equilibrium at the point $Q$, as shown in Figure 1.

Find the length $A P$
Find the length $P Q$

Edexcel M3 2022 January Q2

7 marks Standard +0.8

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-04_479_853_246_607} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} 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 point $A$ which lies above a smooth horizontal table. The particle $P$ moves in a horizontal circle on the table with the string taut. The centre of the circle is the point $O$ on the table, where $A O$ is vertical and the string makes a constant angle $\theta ^ { \circ }$ with $A O$, as shown in Figure 2. Given that $P$ moves with constant angular speed $\sqrt { \frac { 2 g } { a } }$, find the range of possible values of $\theta$

Edexcel M3 2022 January Q3

8 marks Challenging +1.2

A particle $P$ is moving along the $x$-axis. At time $t$ seconds, where $t \geqslant 0 , P$ is $x$ metres from the origin $O$ and is moving with speed $v \mathrm {~ms} ^ { - 1 }$

The acceleration of $P$ has magnitude $\frac { 2 } { ( 2 x + 1 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }$ and is directed towards $O$ When $t = 0 , P$ passes through $O$ in the positive $x$ direction with speed $1 \mathrm {~ms} ^ { - 1 }$

Find $v$ in terms of $x$
Show that $x = \frac { 1 } { 2 } ( \sqrt { ( 4 t + 1 ) } - 1 )$

Edexcel M3 2022 January Q4

11 marks Standard +0.8

A uniform solid hemisphere $H$ has radius $r$ and centre $O$

Show that the centre of mass of $H$ is $\frac { 3 r } { 8 }$ from $O$ $$\left[ \text { You may assume that the volume of } H \text { is } \frac { 2 \pi r ^ { 3 } } { 3 } \right]$$ A uniform solid $S$, shown below in Figure 3, is formed by attaching a uniform solid right circular cylinder of height $h$ and radius $r$ to $H$, so that one end of the cylinder coincides with the plane face of $H$. The point $A$ is the point on $H$ such that $O A = r$ and $O A$ is perpendicular to the plane face of $H$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-12_592_791_909_660} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
Show that the distance of the centre of mass of $S$ from $A$ is $$\frac { 5 r ^ { 2 } + 12 r h + 6 h ^ { 2 } } { 8 r + 12 h }$$ The solid $S$ can rest in equilibrium on a horizontal plane with any point of the curved surface of the hemisphere in contact with the plane.
Find $r$ in terms of $h$.

Edexcel M3 2022 January Q5

13 marks Challenging +1.2

5. \begin{figure}[h]

[diagram]

\captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} 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 $O$. The particle $P$ is held at rest vertically below $O$, with the string taut, as shown in Figure 4. The particle is then projected horizontally with speed $u$, where $u > \sqrt { 2 a g }$ Air resistance is modelled as being negligible.
At the instant when the string makes an angle $\theta$ with the upward vertical through $O$, the speed of $P$ is $v$ and the string goes slack.

Show that $3 v ^ { 2 } = u ^ { 2 } - 2 a g$ From the instant when the string goes slack to the instant when $O P$ is next horizontal, $P$ moves as a projectile. The time from the instant when the string goes slack to the instant when $O P$ is next horizontal is $T$ Given that $\theta = 30 ^ { \circ }$
show that $T = \frac { 2 v } { g }$
Hence, show that the string goes taut again when it is next horizontal.

Edexcel M3 2022 January Q6

15 marks Challenging +1.2

A particle $P$ of mass $m$ is attached to one end of a light elastic spring of natural length 2l. The other end of the spring is attached to a fixed point $A$. The particle $P$ hangs in equilibrium vertically below $A$, at the point $E$ where $A E = 6 l$. The particle $P$ is then raised a vertical distance $2 l$ and released from rest.

Air resistance is modelled as being negligible.

Show that $P$ moves with simple harmonic motion of period $T$ where $$T = 4 \pi \sqrt { \frac { l } { g } }$$
Find, in terms of $m , l$ and $g$, the kinetic energy of $P$ as it passes through $E$
Find, in terms of $T$, the exact time from the instant when $P$ is released to the instant when $P$ has moved a distance 31 .

Edexcel M3 2022 January Q7

15 marks Challenging +1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-24_396_992_246_539} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point $O$ on a rough plane which is inclined to the horizontal at an angle $\alpha$ The string lies along a line of greatest slope of the plane.
The particle $P$ is held at rest on the plane at the point $A$, where $O A = a$, as shown in Figure 5. The particle $P$ is released from $A$ and slides down the plane, coming to rest at the point $B$. The coefficient of friction between $P$ and the plane is $\mu$, where $\mu < \tan \alpha$ Air resistance is modelled as being negligible.

Show that $A B = a ( \sin \alpha - \mu \cos \alpha )$. Given that $\tan \alpha = \frac { 3 } { 4 }$ and $\mu = \frac { 1 } { 2 }$
find, in terms of $a$ and $g$, the maximum speed of $P$ as it moves from $A$ to $B$
Describe the motion of $P$ after it reaches $B$, justifying your answer.

A particle $P$ of mass $m \mathrm {~kg}$ is initially held at rest at the point $O$ on a smooth inclined plane. The plane is inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 2 } { 5 }$

The particle is released from rest and slides down the plane against a force which acts towards $O$. The force has magnitude $\frac { 1 } { 3 } m x ^ { 2 } \mathrm {~N}$, where $x$ metres is the distance of $P$ from $O$.

Find the speed of $P$ when $x = 2$ The particle first comes to instantaneous rest at the point $A$.
Find the distance $O A$.

Edexcel M3 2022 January Q4

8 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bd1e2b07-4a87-49d6-addd-c9f67467ef2f-12_659_513_246_774} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A thin uniform right hollow cylinder, of radius $2 a$ and height $k a$, has a base but no top. A thin uniform hemispherical shell, also of radius $2 a$, is made of the same material as the cylinder. The hemispherical shell is attached to the end of the cylinder forming a container $C$. The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of $C$ is $O$, as shown in Figure 3.

Show that the distance from $O$ to the centre of mass of $C$ is $$\frac { \left( k ^ { 2 } + 4 k + 4 \right) } { 2 ( k + 3 ) } a$$ The container is placed with its circular base on a plane which is inclined at $30 ^ { \circ }$ to the horizontal. The plane is sufficiently rough to prevent $C$ from sliding. The container is on the point of toppling.
Find the value of $k$.

Edexcel M3 2022 January Q5

12 marks Moderate -0.3

A particle $P$ is moving along the $x$-axis. At time $t$ seconds the displacement of $P$ from the origin $O$ is $x$ metres, where $x = 4 \cos \left( \frac { 1 } { 5 } \pi t \right)$

Prove that $P$ is moving with simple harmonic motion.
Find the period of the motion.
State the amplitude of the motion.
Find, in terms of $\pi$, the maximum speed of $P$ The points $A$ and $B$ lie on the $x$-axis, on opposite sides of $O$, with $O A = 1.5 \mathrm {~m}$ and $O B = 2.5 \mathrm {~m}$.
Find the time taken by $P$ to move directly from $A$ to $B$.

Edexcel M3 2022 January Q6

15 marks Standard +0.8

A particle $P$ of mass 1.2 kg is attached to the midpoint of a light elastic string of natural length 0.5 m and modulus of elasticity $\lambda$ newtons.

The fixed points $A$ and $B$ are 0.8 m apart on a horizontal ceiling. One end of the string is attached to $A$ and the other end of the string is attached to $B$. Initially $P$ is held at rest at the midpoint $M$ of the line $A B$ and the tension in the string is 30 N .

Show that $\lambda = 50$ The particle is now held at rest at the point $C$, where $C$ is 0.3 m vertically below $M$. The particle is released from rest.
Find the magnitude of the initial acceleration of $P$
Find the speed of $P$ at the instant immediately before it hits the ceiling.

Edexcel M3 2022 January Q7

17 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bd1e2b07-4a87-49d6-addd-c9f67467ef2f-24_518_538_264_753} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light rod of length $l$. The other end of the rod is attached to a fixed point $O$. The rod can rotate freely in a vertical plane about $O$. The particle is projected with speed $u$ from a point $A$. The line $O A$ makes an angle $\alpha$ with the upward vertical through $O$, where $\alpha < \frac { \pi } { 2 }$ When $O P$ makes an angle $\theta$ with the upward vertical through $O$, the speed of $P$ is $v$, as shown in Figure 4.

Show that $v ^ { 2 } = u ^ { 2 } - 2 g l ( \cos \theta - \cos \alpha )$ Given that $\cos \alpha = \frac { 2 } { 5 }$ and that $u = \sqrt { 3 g l }$
show that $P$ moves in a complete vertical circle. As the rod rotates, the least tension in the rod is $T$ and the greatest tension is $k T$
Find the exact value of $k$

Edexcel M3 2023 January Q1

8 marks Standard +0.3

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-02_703_561_280_753} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The shaded region R is bounded by the x -axis, the line with equation $\mathrm { x } = 1$, the curve with equation $y = 1 + \sqrt { x }$ and the y-axis, as shown in Figure 1. The unit of length on both of the axes is 1 m . The region R is rotated through $2 \pi$ radians about the x-axis to form a solid of revolution which is used to model a uniform solid $S$. Show, using the model and algebraic integration, that

the volume of $S$ is $\frac { 17 \pi } { 6 } \mathrm {~m} ^ { 3 }$
the centre of mass of $S$ is $\frac { 49 } { 85 } \mathrm {~m}$ from 0 . \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-02_2264_41_314_1987}

Edexcel M3 2023 January Q2

6 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-04_252_842_285_609} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A light elastic string AB has natural length I and modulus of elasticity 2 mg .
The end A of the elastic string is attached to a fixed point. The other end B is attached to a particle of mass m . The particle is held in equilibrium, with the elastic string taut and horizontal, by a force of magnitude F . The line of action of the force and the elastic string lie in the same vertical plane. The direction of the force makes an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$, with the upward vertical, as shown in Figure 2.
Find, in terms of I , the length AB . \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-04_2264_53_311_1981}

Edexcel M3 2023 January Q3

10 marks Challenging +1.2

\includegraphics[max width=\textwidth, alt={}]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-06_908_1367_269_349}

A square ABCD of side 4a is made from thin uniform cardboard. The centre of the square is 0 . A circle with centre 0 and radius $\frac { 7 a } { 4 }$ is then removed from the square to form a template T, shown shaded in Figure 3.
A right conical shell, with no base, has radius $\frac { 7 a } { 4 }$ and perpendicular height $6 a$.
The shell is made of the same thin uniform cardboard as T.
The shell is attached to T so that the circumference of the end of the shell coincides with the circumference of the circle centre 0 , to form the hat H , shown in Figure 4.
[0pt] [The surface area of a right conical shell of radius r and slant height I is $\pi r l$.]

Show that the exact distance of the centre of mass of H from O is $$\frac { 175 \pi a } { ( 63 \pi + 128 ) }$$ A fixed rough plane is inclined to the horizontal at an angle $\alpha$. The hat H is placed on the plane, with ABCD in contact with the plane, and AB parallel to a line of greatest slope of the plane. The plane is sufficiently rough to prevent the hat from sliding down the plane. Given that the hat is on the point of toppling,
find the exact value of $\tan \alpha$, giving your answer in simplest form.

Edexcel M3 2023 January Q4

10 marks Challenging +1.2

In this question you must show all stages in your working. Solutions relying entirely on calculator technology are not acceptable.

A particle $P$ is moving along the $x$-axis.
At time $t$ seconds, where $0 \leqslant t \leqslant \frac { 2 } { 3 } , P$ is $x$ metres from the origin 0 and is moving with velocity $\mathrm { v } \mathrm { m } \mathrm { s } ^ { - 1 }$ in the positive x direction where $$v = ( 2 x + 1 ) ^ { \frac { 3 } { 2 } }$$ When $\mathrm { t } = 0 , \mathrm { P }$ passes through 0 .

Find the value of x when the acceleration of P is $243 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
Find v in terms of t .

Edexcel M3 2023 January Q5

14 marks Standard +0.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-12_535_674_283_699} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $\mathrm { a } \sqrt { 3 }$. The other end of the string is attached to a fixed point A . The particle P is also attached to one end of a second light inextensible string of length a. The other end of this string is attached to a fixed point B , where B is vertically below A , with $\mathrm { AB } = \mathrm { a }$. The particle $P$ moves in a horizontal circle with centre 0 , where 0 is vertically below $B$.
The particle P moves with constant angular speed $\omega$, with both strings taut, as shown in Figure 5.

Show that the upper string makes an angle of $30 ^ { \circ }$ with the downward vertical and the lower string makes an angle of $60 ^ { \circ }$ with the downward vertical.
Show that the tension in the upper string is $\frac { 1 } { 2 } m \sqrt { 3 } \left( 2 g - a \omega ^ { 2 } \right)$.
Show that $\frac { 2 g } { 3 a } < \omega ^ { 2 } < \frac { 2 g } { a }$
$\_\_\_\_$ VIAV SIHI NI JIIHM ION OC
VILU SIHIL NI GLIUM ION OC
VEYV SIHI NI III HM ION OC

Edexcel M3 2023 January Q6

14 marks Challenging +1.2

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-16_574_506_283_776} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} A small smooth ring $R$ of mass $m$ is threaded on to a smooth wire in the shape of a circle with centre 0 and radius $I$. The wire is fixed in a vertical plane. The ring $R$ is attached to one end of a light elastic string of natural length I and modulus of elasticity mg . The other end of the elastic string is attached to A , the lowest point of the wire. The point B is on the wire and $O B$ is horizontal. The ring $R$ is at rest at the highest point of the wire, as shown in Figure 6.
The ring $R$ is slightly disturbed from rest and slides along the wire.
At the instant when $R$ reaches the point $B$, the speed of $R$ is $v$ and the magnitude of the force exerted on R by the wire is N .

Show that $$v ^ { 2 } = 2 g l \sqrt { 2 }$$
Show that $$N = \frac { 1 } { 2 } m g ( 5 \sqrt { 2 } - 2 )$$
$\_\_\_\_$ VIAV SIHI NI JIIHM ION OC
VILU SIHIL NI GLIUM ION OC
VEYV SIHI NI ELIUM ION OC

Edexcel M3 2023 January Q7

13 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-20_358_1161_278_452} \captionsetup{labelformat=empty} \caption{Figure 7}

\end{figure} Two points $A$ and $B$ lie on a smooth horizontal table where $A B = 41$.
A particle $P$ of mass $m$ is attached to one end of a light elastic spring of natural length I and modulus of elasticity 2 mg . The other end of the spring is attached to A . The particle P is also attached to one end of another light elastic spring of natural length I and modulus of elasticity mg . The other end of the spring is attached to B.
The particle $P$ rests in equilibrium on the table at the point 0 , where $A 0 = \frac { 5 } { 3 } I$, as shown in Figure 7.
The particle $P$ is moved a distance $\frac { 1 } { 2 } \mathrm { I }$ along the table, from 0 towards $A$, and released from rest.

Show that P moves with simple harmonic motion of period T , where $$\mathrm { T } = 2 \pi \sqrt { \frac { l } { 3 g } }$$
Find, in terms of I and g , the speed of P as it passes through 0 .
Find, in terms of g , the maximum acceleration of P .
Find the exact time, in terms of I and g , from the instant when P is released from rest to the instant when P is first moving with speed $\frac { 3 } { 4 } \sqrt { g l }$ \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-20_2269_56_311_1978} $\_\_\_\_$ VIAV SIHI NI JIIHM ION OC
VILU SIHIL NI GLIUM ION OC
VEYV SIHI NI ELIUM ION OC

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