Questions - OCR MEI Further Mechanics Major

OCR MEI Further Mechanics Major 2019 June Q1

1 Three forces represented by the vectors $- 4 \mathbf { i } , \mathbf { i } + 2 \mathbf { j }$ and $k \mathbf { i } - 2 \mathbf { j }$ act at the points with coordinates $( 0,0 ) , ( 3,0 )$ and $( 0,4 )$ respectively.

Given that the three forces form a couple, find the value of $k$.
Find the magnitude and direction of the couple.

OCR MEI Further Mechanics Major 2019 June Q2

2 The Reynolds number, $R$, is an important dimensionless quantity in fluid dynamics; it can be used to predict flow patterns when a fluid is in motion relative to a surface.
The Reynolds number is defined as $\quad R = \frac { \rho u l } { \mu }$,
where $\rho$ is the density of the fluid, $u$ is the velocity of the fluid relative to the surface, $l$ is the distance travelled by the fluid and $\mu$ is the viscosity of the fluid. Find the dimensions of $\mu$.

OCR MEI Further Mechanics Major 2019 June Q3

3 A ball of mass 2 kg is moving with velocity $( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ when it is struck by a bat. The impulse on the ball is $( - 8 \mathbf { i } + 10 \mathbf { j } )$ Ns.

Find the speed of the ball immediately after the impact.
State one modelling assumption you have used in answering part (a). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-03_373_558_315_258} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows a uniform lamina ABCDE such that ABDE is a rectangle and BCD is an isosceles triangle. $\mathrm { AB } = 5 a , \mathrm { AE } = 4 a$ and $\mathrm { BC } = \mathrm { CD }$. The point F is the midpoint of BD and $\mathrm { FC } = a$.
Find, in terms of $a$, the exact distance of the centre of mass of the lamina from AE. The lamina is freely suspended from B and hangs in equilibrium.
Find the angle between AB and the downward vertical.

OCR MEI Further Mechanics Major 2019 June Q5

5 A particle P of mass 4 kilograms moves in such a way that its position vector at time $t$ seconds is r metres, where
$\mathbf { r } = 3 t \mathbf { i } + 2 \mathrm { e } ^ { - 3 t } \mathbf { j }$.

Find the initial kinetic energy of P .
Find the time when the acceleration of P is 2 metres per second squared. Section B (93 marks)

OCR MEI Further Mechanics Major 2019 June Q6

6 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-04_483_828_370_246} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure} The rim of a smooth hemispherical bowl is a circle of centre O and radius $a$. The bowl is fixed with its rim horizontal and uppermost. A particle P of mass $m$ is released from rest at a point A on the rim as shown in Fig. 6. When P reaches the lowest point of the bowl it collides directly with a stationary particle Q of mass $\frac { 1 } { 2 } m$. After the collision Q just reaches the rim of the bowl. Find the coefficient of restitution between P and Q .

OCR MEI Further Mechanics Major 2019 June Q8

8 A car of mass 800 kg travels up a line of greatest slope of a straight road inclined at $5 ^ { \circ }$ to the horizontal. The power developed by the car is constant and equal to 25 kW . The resistance to the motion of the car is constant and equal to 750 N . The car passes through a point A on the road with speed $7 \mathrm {~ms} ^ { - 1 }$.

Find
- the acceleration of the car at A ,
- the greatest steady speed at which the car can travel up the hill.
The car later passes through a point B on the road where $\mathrm { AB } = 131 \mathrm {~m}$. The time taken to travel from A to B is 10.4 s .
Calculate the speed of the car at B. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-06_442_346_292_251} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} A particle P of mass $m$ is joined to a fixed point O by a light inextensible string of length $l . \mathrm { P }$ is released from rest with the string taut and making an acute angle $\alpha$ with the downward vertical, as shown in Fig. 9. At a time $t$ after P is released the string makes an angle $\theta$ with the downward vertical and the tension in the string is $T$. Angles $\alpha$ and $\theta$ are measured in radians.
Show that $$\left( \frac { \mathrm { d } \theta } { \mathrm {~d} t } \right) ^ { 2 } = \frac { 2 g } { l } \cos \theta + k _ { 1 } ,$$ where $k _ { 1 }$ is a constant to be determined in terms of $g , l$ and $\alpha$.
Show that $$T = 3 m g \cos \theta + k _ { 2 } ,$$ where $k _ { 2 }$ is a constant to be determined in terms of $m , g$ and $\alpha$. It is given that $\alpha$ is small enough for $\alpha ^ { 2 }$ to be negligible.
Find, in terms of $m$ and $g$, the approximate tension in the string.
Show that the motion of P is approximately simple harmonic.

OCR MEI Further Mechanics Major 2019 June Q10

10 A particle P , of mass $m$, moves on a rough horizontal table. P is attracted towards a fixed point O on the table by a force of magnitude $\frac { k m g } { x ^ { 2 } }$, where $x$ is the distance OP. The coefficient of friction between P and the table is $\mu$.
P is initially projected in a direction directly away from O . The velocity of P is first zero at a point A which is a distance $a$ from O .

Show that the velocity $v$ of P , when P is moving away from O , satisfies the differential equation $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( v ^ { 2 } \right) + \frac { 2 k g } { x ^ { 2 } } + 2 \mu g = 0$$
Verify that $$v ^ { 2 } = 2 g k \left( \frac { 1 } { x } - \frac { 1 } { a } \right) + 2 \mu g ( a - x )$$
Find, in terms of $k$ and $a$, the range of values of $\mu$ for which P remains at A .

OCR MEI Further Mechanics Major 2019 June Q11

11 Two uniform smooth spheres A and B have equal radii and are moving on a smooth horizontal surface. The mass of $A$ is 0.2 kg and the mass of $B$ is 0.6 kg . The spheres collide obliquely. When the spheres collide the line joining their centres is parallel to $\mathbf { i }$. Immediately before the collision the velocity of A is $\mathbf { u } _ { \mathrm { A } } \mathrm { ms } ^ { - 1 }$ and the velocity of B is $\mathbf { u } _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }$. The coefficient of restitution between A and B is 0.5. Immediately after the collision the velocity of A is $( - 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ and the velocity of B is $( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$.

Find $\mathbf { u } _ { \mathrm { A } }$ and $\mathbf { u } _ { \mathrm { B } }$. After the collision B collides with a smooth vertical wall which is parallel to $\mathbf { j }$.
The loss in kinetic energy of B caused by the collision with the wall is 1.152 J .
Find the coefficient of restitution between B and the wall.
Find the angle through which the direction of motion of B is deflected as a result of the collision with the wall. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-08_730_476_264_251} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} The ends of a light inextensible string are fixed to two points A and B in the same vertical line, with A above B. The string passes through a small smooth ring of mass $m$. The ring is fastened to the string at a point P . When the string is taut the angle APB is a right angle, the angle BAP is $\theta$ and the perpendicular distance of P from AB is $r$. The ring moves in a horizontal circle with constant angular velocity $\omega$ and the string taut as shown in Fig. 12.
By resolving horizontally and vertically, show that the tension in the part of the string BP is $m \left( r \omega ^ { 2 } \cos \theta - g \sin \theta \right)$.
Find a similar expression, in terms of $r , \omega , m , g$ and $\theta$, for the tension in the part of the string AP. It is given that $\mathrm { AB } = 5 a$ and $\mathrm { AP } = 4 a$.
Show that $16 a \omega ^ { 2 } > 5 g$. The ring is now free to move on the string but remains in the same position on the string as before. The string remains taut and the ring continues to move in a horizontal circle.
Find the period of the motion of the ring, giving your answer in terms of $a , g$ and $\pi$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-09_838_1132_280_248} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure} A step-ladder has two sides AB and AC , each of length $4 a$. Side AB has weight $W$ and its centre of mass is at the half-way point; side AC is light. The step-ladder is smoothly hinged at A and the two parts of the step-ladder, AB and AC , are connected by a light taut rope DE , where D is on $\mathrm { AB } , \mathrm { E }$ is on AC and $\mathrm { AD } = \mathrm { AE } = a$. A man of weight $4 W$ stands at a point F on AB , where $\mathrm { BF } = x$.
The system is in equilibrium with B and C on a smooth horizontal floor and the sides AB and AC are each at an angle $\theta$ to the vertical, as shown in Fig. 13.
By taking moments about A for side AB of the step-ladder and then for side AC of the step-ladder show that the tension in the rope is $$W \left( 1 + \frac { 2 x } { a } \right) \tan \theta .$$ The rope is elastic with natural length $\frac { 1 } { 4 } a$ and modulus of elasticity $W$.
Show that the condition for equilibrium is that $$x = \frac { 1 } { 2 } a ( 8 \cos \theta - \cot \theta - 1 ) .$$ \section*{In this question you must show detailed reasoning.}
Hence determine, in terms of $a$, the maximum value of $x$ for which equilibrium is possible.

OCR MEI Further Mechanics Major 2022 June Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-02_645_609_459_246} Three forces of magnitudes $4 \mathrm {~N} , 7 \mathrm {~N}$ and P N act at a point in the directions shown in the diagram. The forces are in equilibrium.

Draw a closed figure to represent the three forces.
Hence, or otherwise, find the following.
1. The value of $\theta$.
2. The value of $P$.

OCR MEI Further Mechanics Major 2022 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-03_359_931_251_255} A particle is projected with speed V from a point O on horizontal ground. The angle of projection is $\theta$ above the horizontal. The particle passes, in succession, through two points A and B , each at a height h above the ground and a distance d apart, as shown in the diagram. You are given that $d ^ { 2 } = \frac { v ^ { \alpha } \sin ^ { 2 } 2 \theta } { g ^ { \beta } } - \frac { 8 h v ^ { 2 } \cos ^ { 2 } \theta } { g }$. Use dimensional analysis to find $\alpha$ and $\beta$.

OCR MEI Further Mechanics Major 2022 June Q3

3 A particle, of mass 2 kg , is placed at a point A on a rough horizontal surface. There is a straight vertical wall on the surface and the point on the wall nearest to $A$ is $B$. The distance $A B$ is 5 m . The particle is projected with speed $4.2 \mathrm {~ms} ^ { - 1 }$ along the surface from A towards B . The particle hits the wall directly and rebounds. The coefficient of friction between the particle and the surface is 0.1 .

Determine the speed of the particle immediately before impact with the wall. The magnitude of the impulse that the wall exerts on the particle is 9.8 Ns .
Find the speed of the particle immediately after impact with the wall.

OCR MEI Further Mechanics Major 2022 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-04_629_835_260_251} The diagram shows a particle P , of mass 0.1 kg , which is attached by a light inextensible string of length 0.5 m to a fixed point O . P moves with constant angular speed $5 \mathrm { rad } \mathrm { s } ^ { - 1 }$ in a horizontal circle with centre vertically below O . The string is inclined at an angle $\theta$ to the vertical.

Determine the tension in the string.
Find the value of $\theta$.
Find the kinetic energy of P.

OCR MEI Further Mechanics Major 2022 June Q5

5 At time $t$ seconds, where $t \geqslant 0$, a particle P of mass 2 kg is moving on a smooth horizontal surface. The particle moves under the action of a constant horizontal force of ( $- 2 \mathbf { i } + 6 \mathbf { j }$ ) N and a variable horizontal force of $( 2 \cos 2 t \mathbf { i } + 4 \sin t \mathbf { j } ) \mathrm { N }$. The acceleration of P at time t seconds is $\mathrm { am } \mathrm { S } ^ { - 2 }$.

Find a in terms of t. The particle P is at rest when $\mathrm { t } = 0$.
Determine the speed of P at the instant when $\mathrm { t } = 2$. Answer all the questions.
Section B (91 marks)

OCR MEI Further Mechanics Major 2022 June Q7

7 Two small uniform smooth spheres A and B , of masses 2 kg and 3 kg respectively, are moving in opposite directions along the same straight line towards each other on a smooth horizontal surface. Sphere $A$ has speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and B has speed $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ before they collide. The coefficient of restitution between A and B is e .

Show that the velocity of B after the collision, in the original direction of motion of A , is $\frac { 1 } { 5 } ( 1 + 6 e ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and find a similar expression for the velocity of A after the collision.
The following three parts are independent of each other, and each considers a different scenario regarding the collision between A and B .
1. In the collision between A and B the spheres coalesce to form a combined body C . State the speed of C after the collision.
2. In the collision between A and B the direction of motion of A is reversed. Find the range of possible values of e .
3. The total loss in kinetic energy due to the collision is 3 J . Determine the value of e.

OCR MEI Further Mechanics Major 2022 June Q8

8 A particle P is projected from a fixed point O with initial velocity $\mathbf { u i } + \mathrm { kuj }$, where k is a positive constant. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in the horizontal and vertically upward directions respectively. $P$ moves with constant gravitational acceleration of magnitude $g$. At time $t \geqslant 0$, particle $P$ has position vector $\mathbf { r }$ relative to $O$.

Starting from an expression for $\ddot { \mathbf { r } }$, use integration to derive the formula $$\mathbf { r } = u \mathbf { t } + \left( k u t - \frac { 1 } { 2 } g t ^ { 2 } \right) \mathbf { j } .$$ The position vector $\mathbf { r }$ of P at time $\mathrm { t } \geqslant 0$ can be expressed as $\mathbf { r } = x \mathbf { i } + y \mathbf { j }$, where the axes Ox and Oy are horizontally and vertically upwards through O respectively. The axis Ox lies on horizontal ground.
Show that the path of P has cartesian equation $$g x ^ { 2 } - 2 k u ^ { 2 } x + 2 u ^ { 2 } y = 0 .$$
Hence find, in terms of $\mathrm { g } , \mathrm { k }$ and u , the maximum height of P above the ground during its motion. The maximum height P reaches above the ground is equal to the distance OA , where A is the point where P first hits the ground.
Determine the value of k .

OCR MEI Further Mechanics Major 2022 June Q9

9 [In this question you may use the facts that for a uniform solid right circular cone of height $h$ and base radius $r$ the volume is $\frac { 1 } { 3 } \pi r ^ { 2 } h$ and the centre of mass is $\frac { 1 } { 4 } h$ above the base on the line from the centre of the base to the vertex.]
\includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-07_677_963_395_248} The diagram shows the shaded region S bounded by the curve $\mathrm { y } = \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { x } }$ for $0 \leqslant x \leqslant 2$, the x -axis, the y -axis, and the line $\mathrm { y } = \frac { 1 } { 4 } \mathrm { e } ( 6 - \mathrm { x } )$. The line $\mathrm { y } = \frac { 1 } { 4 } \mathrm { e } ( 6 - \mathrm { x } )$ meets the curve $\mathrm { y } = \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { x } }$ at the point A with coordinates (2,e).
The region S is rotated through $2 \pi$ radians about the x -axis to form a uniform solid of revolution T .

Show that the x -coordinate of the centre of mass of T is $\frac { 3 \left( 5 \mathrm { e } ^ { 2 } + 1 \right) } { 7 \mathrm { e } ^ { 2 } - 3 }$. Solid T is freely suspended from A and hangs in equilibrium.
Determine the angle between AO , where O is the origin, and the vertical.
\includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-08_725_1541_251_248} A small toy car runs along a track in a vertical plane.
The track consists of three sections: a curved section AB , a horizontal section BC which rests on the floor, and a circular section that starts at C with centre O and radius r m . The section BC is tangential to the curved section at B and tangential to the circular section at C , as shown in the diagram. The car, of mass mkg , is placed on the track at A , at a height hm above the floor, and released from rest. The car runs along the track from A to C and enters the circular section at C . It can be assumed that the track does not obstruct the car moving on to the circular section at C . The track is modelled as being smooth, and the car is modelled as a particle P .
Show that, while P remains in contact with the circular section of the track, the magnitude of the normal contact force between P and the circular section is
$\mathrm { mg } \left( 3 \cos \theta - 2 + \frac { 2 \mathrm {~h} } { \mathrm { r } } \right) \mathrm { N }$,
where $\theta$ is the angle between OC and OP .
Hence determine, in terms of r , the least possible value of h so that P can complete a vertical circle.
Apart from not modelling the car as a particle, state one refinement that would make the model more realistic.
\includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-09_668_695_258_251} The diagram shows two small identical uniform smooth spheres, A and B, just before A collides with B. Sphere B is at rest on a horizontal table with its centre vertically above the edge of the table. Sphere A is projected vertically upwards so that, just before it collides with B, the speed of A is $\mathrm { U } \mathrm { m } \mathrm { s } ^ { - 1 }$ and it is in contact with the vertical side of the table. The point of contact of A with the vertical side of the table and the centres of the spheres are in the same vertical plane.
Show that on impact the line of centres makes an angle of $30 ^ { \circ }$ with the vertical. The coefficient of restitution between A and B is $\frac { 1 } { 2 }$. After the impact B moves freely under gravity.
Determine, in terms of U and g , the time taken for B to first return to the table.
\includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-10_867_1045_255_255} The diagram shows a uniform square lamina ABCD , of weight W and side-length $a$. The lamina is in equilibrium in a vertical plane that also contains the point O . The vertex A rests on a smooth plane inclined at an angle of $30 ^ { \circ }$ to the horizontal. The vertex B rests on a smooth plane inclined at an angle of $60 ^ { \circ }$ to the horizontal. OA is a line of greatest slope of the plane inclined at $30 ^ { \circ }$ to the horizontal and OB is a line of greatest slope of the plane inclined at $60 ^ { \circ }$ to the horizontal. The side AB is inclined at an angle $\theta$ to the horizontal and the lamina is kept in equilibrium in this position by a clockwise couple of magnitude $\frac { 1 } { 8 } \mathrm { aW }$.
By resolving horizontally and vertically, determine, in terms of W, the magnitude of the normal contact force between the plane and the lamina at B .
By taking moments about A , show that $\theta$ satisfies the equation $$2 ( \sqrt { 3 } + 2 ) \sin \theta - 2 \cos \theta = 1 .$$
Verify that $\theta = 22.4 ^ { \circ }$, correct to 1 decimal place.

OCR MEI Further Mechanics Major 2023 June Q1

1 A car of mass 800 kg moves in a straight line along a horizontal road.
There is a constant resistance to the motion of the car of magnitude 600 N .
When the car is travelling at a speed of $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the power developed by the car is 27 kW .
Determine the acceleration of the car when it is travelling at $15 \mathrm {~ms} ^ { - 1 }$.

OCR MEI Further Mechanics Major 2023 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-02_232_609_840_239} Two small uniform smooth spheres A and B have masses 0.5 kg and 2 kg respectively. The two spheres are travelling in the same direction in the same straight line on a smooth horizontal surface. Sphere $A$ is moving towards $B$ with speed $6 \mathrm {~ms} ^ { - 1 }$ and $B$ is moving away from $A$ with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see diagram). Spheres A and B collide. After this collision A moves with speed $0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Determine the possible speeds with which B moves after the collision.

OCR MEI Further Mechanics Major 2023 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-03_565_757_251_242} The diagram shows a particle P , of mass 0.2 kg , which is attached by a light inextensible string of length 0.75 m to a fixed point O . Particle P moves with constant angular speed $\omega$ rads $^ { - 1 }$ in a horizontal circle with centre vertically below O . The string is inclined at an angle $\theta$ to the vertical. The greatest tension that the string can withstand without breaking is 15 N .

Find the greatest possible value of $\theta$, giving your answer to the nearest degree.
Determine the greatest possible value of $\omega$.

OCR MEI Further Mechanics Major 2023 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-04_598_696_255_246}
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-04_465_67_294_1023} A rigid lamina of negligible mass is in the form of a rhombus ABCD , where $\mathrm { AC } = 6 \mathrm {~m}$ and $\mathrm { BD } = 8 \mathrm {~m}$. Forces of magnitude $2 \mathrm {~N} , 4 \mathrm {~N} , 3 \mathrm {~N}$ and 5 N act along its sides $\mathrm { AB } , \mathrm { BC } , \mathrm { CD }$ and DA , respectively, as shown in the diagram. A further force $\mathbf { F }$ N, acting at A, and a couple of magnitude $G N m$ are also applied to the lamina so that it is in equilibrium.

Determine the magnitude and direction of $\mathbf { F }$.
Determine the value of $G$.

OCR MEI Further Mechanics Major 2023 June Q5

5 A particle P of mass $m \mathrm {~kg}$ is projected with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a rough horizontal surface. During the motion of P , a constant frictional force of magnitude $F \mathrm {~N}$ acts on P . When the velocity of P is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, it experiences a force of magnitude $k v \mathrm {~N}$ due to air resistance, where $k$ is a constant.

Determine the dimensions of $k$. At time $T$ s after projection P comes to rest. A formula approximating the value of $T$ is
$\mathrm { T } = \frac { \mathrm { mu } } { \mathrm { F } } - \frac { \mathrm { kmu } ^ { 2 } } { 2 \mathrm {~F} ^ { 2 } } + \frac { 1 } { 3 } \mathrm { k } ^ { 2 } \mathrm {~m} ^ { \alpha } \mathrm { u } ^ { \beta } \mathrm { F } ^ { \gamma }$.
Use dimensional analysis to find $\alpha , \beta$ and $\gamma$.

OCR MEI Further Mechanics Major 2023 June Q6

6 At time $t$ seconds, where $t \geqslant 0$, a particle P has position vector $\mathbf { r }$ metres, where
$\mathbf { r } = \left( 2 t ^ { 2 } - 12 t + 6 \right) \mathbf { i } + \left( t ^ { 3 } + 3 t ^ { 2 } - 8 t \right) \mathbf { j }$.
The velocity of P at time $t$ seconds is $\mathbf { v } \mathrm { ms } ^ { - 1 }$.

Find $\mathbf { v }$ in terms of $t$.
Determine the speed of P at the instant when it is moving parallel to the vector $\mathbf { i } - 4 \mathbf { j }$.
Determine the value of $t$ when the magnitude of the acceleration of P is $20.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

OCR MEI Further Mechanics Major 2023 June Q7

7 One end of a rope is attached to a block A of mass 2 kg . The other end of the rope is attached to a second block B of mass 4 kg . Block A is held at rest on a fixed rough ramp inclined at $30 ^ { \circ }$ to the horizontal. The rope is taut and passes over a small smooth pulley P which is fixed at the top of the ramp. The part of the rope from A to P is parallel to a line of greatest slope of the ramp. Block B hangs vertically below P , at a distance $d \mathrm {~m}$ above the ground, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-05_351_851_1274_244} Block A is more than $d \mathrm {~m}$ from P . The blocks are released from rest and A moves up the ramp. The coefficient of friction between A and the ramp is $\frac { 1 } { 2 \sqrt { 3 } }$. The blocks are modelled as particles, the rope is modelled as light and inextensible, and air resistance can be ignored.

Determine, in terms of $g$ and $d$, the work done against friction as A moves $d \mathrm {~m}$ up the ramp.
Given that the speed of B immediately before it hits the ground is $1.75 \mathrm {~ms} ^ { - 1 }$, use the work-energy principle to determine the value of $d$.
Suggest one improvement, apart from including air resistance, that could be made to the model to make it more realistic.

OCR MEI Further Mechanics Major 2023 June Q8

8
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-06_652_716_258_230} The diagram shows the shaded region R bounded by the curve $\mathrm { y } = \sqrt { 3 \mathrm { x } + 4 }$, the $x$-axis, the $y$-axis, and the straight line that passes through the points $( k , 0 )$ and $( 4,4 )$, where $0 < k < 4$. Region R is occupied by a uniform lamina.

Determine, in terms of $k$, an expression for the $y$-coordinate of the centre of mass of the lamina. Give your answer in the form $\frac { \mathrm { a } + \mathrm { bk } } { \mathrm { c } + \mathrm { dk } }$, where $a , b , c$ and $d$ are integers to be determined.
Show that the $y$-coordinate of the centre of mass of the lamina cannot be $\frac { 3 } { 2 }$.

OCR MEI Further Mechanics Major 2023 June Q11

11
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-09_716_703_251_246} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a right-angled triangle ABC , with AB perpendicular to AC , which lies in a vertical plane. The length of AB is 3 cm , and the length of AC is 12 cm . The prism is resting in equilibrium on a horizontal surface and against a vertical wall. The side AC of the prism makes an angle $\theta$ with the horizontal. A horizontal force of magnitude $P \mathrm {~N}$ is now applied to the prism at B . This force acts towards the wall in the vertical plane which passes through the centre of mass G of the prism and is perpendicular to the wall. The weight of the prism is 15 N and the coefficients of friction between the prism and the surface, and between the prism and the wall, are each $\frac { 1 } { 2 }$.

Show that the least value of $P$ needed to move the prism is given by $$P = \frac { 40 \cos \theta + 95 \sin \theta } { 16 \sin \theta - 13 \cos \theta } .$$
Determine the range in which the value of $\theta$ must lie.

Questions — OCR MEI Further Mechanics Major (73 questions)

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