Questions - SPS SPS FM Mechanics

By taking moments, find the smallest value of $\theta$ for which the ladder is in equilibrium.
A man of mass 75 kg stands on the ladder when $\theta = 60 ^ { \circ }$. Find the greatest distance from $A$ that he can stand without the ladder slipping.
[0pt] [Question 4 Continued]

SPS SPS FM Mechanics 2023 January Q5

5. Two smooth circular discs $A$ and $B$ are moving on a horizontal plane. The masses of $A$ and $B$ are 3 kg and 4 kg respectively. At the instant before they collide

the velocity of $A$ is $2 \mathrm {~ms} ^ { - 1 }$ at an angle of $60 ^ { \circ }$ to the line joining their centres,
the velocity of $B$ is $5 \mathrm {~ms} ^ { - 1 }$ towards $A$ along the line joining their centres (see Fig. 6).

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{15f4500a-8eb8-4b5f-896c-de730272a35b-12_451_961_406_255} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure} Given that the velocity of $A$ after the collision is perpendicular to the velocity of $A$ before the collision find the coefficient of restitution between $A$ and $B$.
[0pt] [Question 5 Continued] \section*{6.}

SPS SPS FM Mechanics 2023 January Q6

6. A light elastic string of natural length $a$ has modulus of elasticity $k m g$, where $k$ is a constant. One end of the string is attached to a fixed point $O$ and the other end is attached to a particle of mass $m$. The particle moves, with the string stretched, in a horizontal circle with constant angular speed $\omega$, with the centre of the circle vertically below $O$.

Show that, if the string makes a constant angle $\theta$ with the vertical, $$\cos \theta = \frac { k g - a \omega ^ { 2 } } { k a \omega ^ { 2 } }$$
Show that $\omega < \sqrt { \frac { k g } { a } }$
[0pt] [Question 6 Continued] Spare space for extra working Spare space for extra working Spare space for extra working Spare space for extra working
[0pt] [End of Examination]

SPS SPS FM Mechanics 2024 January Q3

3.
\includegraphics[max width=\textwidth, alt={}, center]{5f9a87c6-2255-4178-ab04-441bb0cc4ce0-06_397_878_159_571} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses 4 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The spheres are moving in opposite directions, each at $60 ^ { \circ }$ to the line of centres (see diagram). After the collision $A$ moves in a direction perpendicular to the line of centres.

Show that the speed of $B$ is unchanged as a result of the collision, and find the angle that the new direction of motion of $B$ makes with the line of centres.
Find the coefficient of restitution between the spheres.
[0pt] [Question 3 Continued]

SPS SPS FM Mechanics 2024 January Q4

4. A uniform heavy lamina occupies the region shaded in Fig. 3. This region is formed by removing a square of side 1 unit from a square of side $a$ units (where $a > 1$ ). \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5f9a87c6-2255-4178-ab04-441bb0cc4ce0-08_558_594_299_699} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Relative to the axes shown in Fig. 3, the centre of mass of the lamina is at $( \bar { x } , \bar { y } )$.

Show that $\bar { x } = \bar { y } = \frac { a ^ { 2 } + a + 1 } { 2 ( a + 1 ) }$.
[0pt] [You may need to use the result $\frac { a ^ { 3 } - 1 } { 2 \left( a ^ { 2 } - 1 \right) } = \frac { a ^ { 2 } + a + 1 } { 2 ( a + 1 ) }$.]
Show that the centre of mass of the lamina lies on its perimeter if $a = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )$.
With the value of $a = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )$ the lamina is suspended from A and hangs in equilibrium. Find the angle that the line OA makes with the vertical.
[0pt] [Question 4 Continued] \section*{5.}

SPS SPS FM Mechanics 2024 January Q5

5. A cone of semi-vertical angle $60 ^ { \circ }$ is fixed with its axis vertical and vertex upwards. A particle of mass $m$ is attached to one end of a light inextensible string of length $l$. The other end of the string is attached to a fixed point vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed $\omega$, with the string making a constant angle $60 ^ { \circ }$ with the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5f9a87c6-2255-4178-ab04-441bb0cc4ce0-10_538_648_456_664}

Find the tension in the string, in terms of $m , l , \omega$ and $g$. The particle remains on the surface of the cone.
Show that the time for the particle to make one complete revolution is greater than $$2 \pi \sqrt { \frac { l \sqrt { 3 } } { 2 g } } .$$ [Question 5 Continued]
[0pt] [Question 5 Continued]

SPS SPS FM Mechanics 2025 January Q1

1. A smooth uniform sphere $A$, of mass $5 m$ and radius $r$, is at rest on a smooth horizontal plane. A second smooth uniform sphere $B$, of mass $3 m$ and radius $r$, is moving in a straight line on the plane with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and strikes $A$. Immediately before the impact the direction of motion of $B$ makes an angle of $60 ^ { \circ }$ with the line of centres of the spheres. The direction of motion of $B$ is turned through an angle of $30 ^ { \circ }$ by the impact. Find

the speed of $B$ immediately after the impact,
the coefficient of restitution between the spheres.

SPS SPS FM Mechanics 2025 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7ecb33af-c165-4b72-98f2-7574fad3fdd5-04_506_613_246_671} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $2 a$ and modulus of elasticity 6 mg . The other end of the string is attached to a fixed point $A$. The particle moves with constant speed $v$ in a horizontal circle with centre $O$, where $O$ is vertically below $A$ and $O A = 2 a$, as shown in Figure 2.

Show that the extension in the string is $\frac { 2 } { 5 } a$.
Find $v ^ { 2 }$ in terms of $a$ and $g$.
[0pt] [Question 2 Continued]

SPS SPS FM Mechanics 2025 January Q3

3. A particle $P$ of mass $2 m$ is moving in a straight line with speed $3 u$ on a smooth horizontal table. A second particle $Q$ of mass $3 m$ is moving in the opposite direction to $P$ along the same straight line with speed $u$. The particle $P$ collides directly with $Q$. The direction of motion of $P$ is reversed by the collision. The coefficient of restitution between $P$ and $Q$ is $e$.

Show that the speed of $Q$ immediately after the collision is $\frac { u } { 5 } ( 8 e + 3 )$
Find the range of possible values of $e$. The total kinetic energy of the particles before the collision is $T$. The total kinetic energy of the particles after the collision is $k T$. Given that $e = \frac { 1 } { 2 }$
find the value of $k$.
[0pt] [Question 3 Continued]

SPS SPS FM Mechanics 2025 January Q4

4. One end $A$ of a light elastic string $A B$, of modulus of elasticity $m g$ and natural length $a$, is fixed to a point on a rough plane inclined at an angle $\theta$ to the horizontal. The other end $B$ of the string is attached to a particle of mass $m$ which is held at rest on the plane. The string $A B$ lies along a line of greatest slope of the plane, with $B$ lower than $A$ and $A B = a$. The coefficient of friction between the particle and the plane is $\mu$, where $\mu < \tan \theta$. The particle is released from rest.

Show that when the particle comes to rest it has moved a distance $2 a ( \sin \theta - \mu \cos \theta )$ down the plane.
Given that there is no further motion, show that $\mu \geqslant \frac { 1 } { 3 } \tan \theta$.
[0pt] [Question 4 Continued]

SPS SPS FM Mechanics 2025 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7ecb33af-c165-4b72-98f2-7574fad3fdd5-10_881_1301_173_397} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform solid right circular cylinder has height $h$ and radius $r$. The centre of one plane face is $O$ and the centre of the other plane face is $Y$. A cylindrical hole is made by removing a solid cylinder of radius $\frac { 1 } { 4 } r$ and height $\frac { 1 } { 4 } h$ from the end with centre $O$. The axis of the cylinder removed is parallel to $O Y$ and meets the end with centre $O$ at $X$, where $O X = \frac { 1 } { 4 } r$. One plane face of the cylinder removed coincides with the plane face through $O$ of the original cylinder. The resulting solid $S$ is shown in Figure 3.

Show that the centre of mass of $S$ is at a distance $\frac { 85 h } { 168 }$ from the plane face
containing $O$. containing $O$. The solid $S$ is freely suspended from $O$. In equilibrium the line $O Y$ is inclined at an angle $\arctan ( 17 )$ to the horizontal.
Find $r$ in terms of $h$.
[0pt] [Question 5 Continued]
[0pt] [Question 5 Continued]

SPS SPS FM Mechanics 2026 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b3e54459-0d05-4858-8978-60fe3d4d1719-04_501_693_242_635} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The uniform L-shaped lamina $O A B C D E$, shown in Figure 2, is made from two identical rectangles. Each rectangle is 4 metres long and $a$ metres wide. Giving each answer in terms of $a$, find the distance of the centre of mass of the lamina from

$O E$,
$O A$. The lamina is freely suspended from $O$ and hangs in equilibrium with $O E$ at an angle $\theta$ to the downward vertical through $O$, where $\tan \theta = \frac { 4 } { 3 }$.
Find the value of $a$.
[0pt] [Question 2 Continued]

SPS SPS FM Mechanics 2026 January Q3

3.
\includegraphics[max width=\textwidth, alt={}, center]{b3e54459-0d05-4858-8978-60fe3d4d1719-06_534_533_191_717} A light elastic string has natural length $8 a$ and modulus of elasticity $5 m g$. A particle $P$ of mass $m$ is attached to the midpoint of the string. The ends of the string are attached to points $A$ and $B$ which are a distance $12 a$ apart on a smooth horizontal table. The particle $P$ is held on the table so that $A P = B P = L$ (see diagram). The particle $P$ is released from rest. When $P$ is at the midpoint of $A B$ it has speed $\sqrt { 80 a g }$.

Find $L$ in terms of $a$.
Find the initial acceleration of $P$ in terms of $g$.
[0pt] [Question 3 Continued] \section*{4.} A hollow hemispherical bowl of radius $a$ has a smooth inner surface and is fixed with its axis vertical. A particle $P$ of mass $m$ moves in horizontal circles on the inner surface of the bowl, at a height $x$ above the lowest point of the bowl. The speed of $P$ is $\sqrt { \frac { 8 } { 3 } g a }$. Find $x$ in terms of $a$.
[0pt] [Question 4 Continued]

SPS SPS FM Mechanics 2026 January Q5

5.
\includegraphics[max width=\textwidth, alt={}, center]{b3e54459-0d05-4858-8978-60fe3d4d1719-10_478_828_178_575}
$A B$ and $B C$ are two fixed smooth vertical barriers on a smooth horizontal surface, with angle $A B C = 60 ^ { \circ }$. A particle of mass $m$ is moving with speed $u$ on the surface. The particle strikes $A B$ at an angle $\theta$ with $A B$. It then strikes $B C$ and rebounds at an angle $\beta$ with $B C$ (see diagram). The coefficient of restitution between the particle and each barrier is $e$ and $\tan \theta = 2$. The kinetic energy of the particle after the first collision is $40 \%$ of its kinetic energy before the first collision.

Find the value of $e$.
Find the size of angle $\beta$.
[0pt] [Question 5 Continued] \section*{6.}
\includegraphics[max width=\textwidth, alt={}]{b3e54459-0d05-4858-8978-60fe3d4d1719-12_511_1145_296_452}
A particle $P$ of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m . The other end of the string is attached to a fixed point $O$. A particle $Q$ of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to $P$. The particle $P$ moves in a horizontal circle of radius 0.8 m with angular speed $\omega \operatorname { rad~s } ^ { - 1 }$. The particle $Q$ moves in a horizontal circle of radius 1.4 m also with angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$. The centres of the circles are vertically below $O$, and $O , P$ and $Q$ are always in the same vertical plane. The strings $O P$ and $P Q$ remain at constant angles $\alpha$ and $\beta$ respectively to the vertical (see diagram).
Find the tension in the string $O P$.
Find the value of $\omega$.
Find the value of $\beta$.
[0pt] [Question 6 Continued]

SPS SPS FM Mechanics 2026 January Q7

7.
\includegraphics[max width=\textwidth, alt={}, center]{b3e54459-0d05-4858-8978-60fe3d4d1719-14_371_880_191_589} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $\frac { 1 } { 2 } m$ respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere $A$ is travelling with speed $u$ and its direction of motion makes an angle $\alpha$ with the line of centres. Sphere $B$ is travelling with speed $2 u$ and its direction of motion makes an angle $\beta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac { 5 } { 8 }$ and $\alpha + \beta = 90 ^ { \circ }$.

Find the component of the velocity of $B$ parallel to the line of centres after the collision, giving your answer in terms of $u$ and $\alpha$. The direction of motion of $B$ after the collision is parallel to the direction of motion of $A$ before the collision.
Find the value of $\tan \alpha$.
[0pt] [Question 7 Continued]

SPS SPS FM Mechanics 2026 January Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b3e54459-0d05-4858-8978-60fe3d4d1719-16_286_933_201_459} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre $O$ and radius $5 a$.
A uniform rod $A B$, of length $16 a$ and weight $W$, rests in equilibrium on the hemisphere with end $A$ on the ground. The rod rests on the hemisphere at the point $C$, where $A C = 12 a$ and angle $C A O = \alpha$, as shown in Figure 1. Points $A , C , B$ and $O$ all lie in the same vertical plane.

Explain why $A O = 13 a$ The normal reaction on the rod at $C$ has magnitude $k W$
Show that $k = \frac { 8 } { 13 }$ The resultant force acting on the rod at $A$ has magnitude $R$ and acts upwards at $\theta ^ { \circ }$ to the horizontal.
Find
1. an expression for $R$ in terms of $W$
2. the value of $\theta$
  (8)
  [0pt] [Question 8 Continued] Spare space for extra working Spare space for extra working Spare space for extra working

SPS SPS FM Mechanics 2022 January Q2

2.
\includegraphics[max width=\textwidth, alt={}, center]{069a48ca-5453-4549-8a9a-0b0eeb2f08af-08_662_540_376_742} A uniform solid right circular cone has base radius $a$ and semi-vertical angle $\alpha$, where $\tan \alpha = \frac { 1 } { 3 }$. The cone is freely suspended by a string attached at a point A on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of $\theta ^ { 0 }$ with the upward vertical, as shown in the diagram above. Find, to one decimal place, the value of $\theta$.
[0pt] [Question 2 Continued]
[0pt] [Question 2 Continued]
[0pt] [Question 2 Continued]

SPS SPS FM Mechanics 2022 January Q3

3. A car of mass 800 kg is driven with its engine generating a power of 15 kW .

The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds.
The car is next driven at constant speed up a straight road inclined at an angle $\theta$ to the horizontal. The resistance to motion is now modelled as being constant with magnitude of 150 N. Given that $\sin \theta = \frac { 1 } { 20 }$, find the speed of the car.
The car is now driven at a constant speed of $30 \mathrm {~ms} ^ { - 1 }$ along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming the resistance to motion of the car is three times the resistance to motion of the trailer. Find:
1. the resistance to motion of the car,
2. the magnitude of the tension in the towbar
  [0pt] [Question 3 Continued]
  [0pt] [Question 3 Continued]
  [0pt] [Question 3 Continued]

SPS SPS FM Mechanics 2022 January Q4

4.
\includegraphics[max width=\textwidth, alt={}, center]{069a48ca-5453-4549-8a9a-0b0eeb2f08af-16_357_840_445_552} Two uniform smooth spheres $A$ and $B$ of equal radius are moving on a horizontal surface when they collide. $A$ has mass 0.1 kg and B has mass 0.4 kg . Immediately before the collision $A$ is moving with speed $2.8 \mathrm {~ms} ^ { - 1 }$ along the line of centres, and $B$ is moving with speed $1 \mathrm {~ms} ^ { - 1 }$ at an angle $\theta$ to the line of centres, where $\cos \theta = 0.8$ (see diagram). Immediately after the collision $A$ is stationary. Find:

the coefficient of restitution between $A$ and $B$,
the angle turned through by the direction of motion of B as a result of the collision.
[0pt] [Question 4 Continued]
[0pt] [Question 4 Continued]
[0pt] [Question 4 Continued] \section*{5.} A right circular cone $C$ of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of $C$. The other end of the string is attached to a particle $P$ of mass 2.5 kg .
$P$ moves in a horizontal circle with constant speed and in contact with the smooth curved surface of $C$. The extension of the string is 1.5 m .
Find the tension in the string.
Find the speed of $P$.
[0pt] [Question 5 Continued]
[0pt] [Question 5 Continued]
[0pt] [Question 5 Continued]

SPS SPS FM Mechanics 2022 January Q6

8 marks

6. A uniform rod, PQ, of length $2 a$, rests with one end, $P$, on rough horizontal ground and a point $T$ resting on a rough fixed prism of semi-circular cross-section of radius $a$, as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both $P$ and $T$ is $\mu$.
\includegraphics[max width=\textwidth, alt={}, center]{069a48ca-5453-4549-8a9a-0b0eeb2f08af-24_531_1291_577_331} The rod is on the point of slipping when it is inclined at an angle of $30 ^ { \circ }$ to the horizontal.
Find the value of $\mu$.
[0pt] [8]
[0pt] [Question 6 Continued]
[0pt] [Question 6 Continued]
[0pt] [Question 6 Continued]

SPS SPS FM Mechanics 2022 January Q7

7. The diagram shows the cross-section through the centre of mass of a uniform solid prism. The crosssection is a trapezium $A B C D$ with $A B$ and $C D$ perpendicular to $A D$. The lengths of $A B$ and $A D$ are each 5 cm and the length of $C D$ is $( a + 5 ) \mathrm { cm }$.
\includegraphics[max width=\textwidth, alt={}, center]{069a48ca-5453-4549-8a9a-0b0eeb2f08af-28_391_640_500_699}

Show the distance of the centre of mass of the prism from $A D$ is $$\frac { a ^ { 2 } + 15 a + 75 } { 3 ( a + 10 ) } \mathrm { cm } .$$ The prism is placed with the face containing $A B$ in contact with a horizontal surface.
Find the greatest value of $a$ for which the prism does not topple. The prism is now placed on an inclined plane which makes an angle $\theta ^ { o }$ with the horizontal. $A B$ lies along a line of greatest slope with $B$ higher than $A$.
Using the value for $a$ found in part (ii), and assuming the prism does not slip down the plane, find the great value of $\theta$ for which the prism does not topple.
[0pt] [Question 7 Continued]
[0pt] [Question 7 Continued]
[0pt] [Question 7 Continued] \footnotetext{[End of Examination] }

Questions — SPS SPS FM Mechanics (46 questions)

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