Questions - OCR M3 - A-Level Maths

Show that $\cos T O B = \frac { \sqrt { 3 } } { 3 }$.
Find the speed of the particle at $B$.
Find the transverse acceleration of the particle at $B$.

OCR M3 2013 June Q6

6 Two uniform rods $A B$ and $B C$, each of length $2 l$, are freely jointed at $B$. The weight of $A B$ is $W$ and the weight of $B C$ is $2 W$. The rods are in a vertical plane with $A$ freely pivoted at a fixed point and $C$ resting in equilibrium on a rough horizontal plane. The normal and frictional components of the force acting on $B C$ at $C$ are $R$ and $F$ respectively. The rod $A B$ makes an angle $30 ^ { \circ }$ to the horizontal and the rod $B C$ makes an angle $60 ^ { \circ }$ to the horizontal (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-4_682_901_479_587}

By considering the equilibrium of $\operatorname { rod } B C$, show that $W + \sqrt { 3 } F = R$.
By taking moments about $A$ for the equilibrium of the whole system, find another equation involving $W , F$ and $R$.
Given that the friction at $C$ is limiting, calculate the value of the coefficient of friction at $C$.

OCR M3 2013 June Q7

7 A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 39.2 mN . The other end of the string is attached to a fixed point $O$. The particle is released from rest at $O$.

Show that, while the string is in tension, the particle performs simple harmonic motion about a point 1 m below $O$.
Show that when $P$ is at its lowest point the extension of the string is 0.8 m .
Find the time after its release that $P$ first reaches its lowest point.
Find the velocity of $P 0.8 \mathrm {~s}$ after it is released from $O$. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR M3 2014 June Q1

1 A particle $P$ of mass 0.3 kg is moving on a smooth horizontal surface with speed $0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it is struck by a horizontal impulse. The magnitude of the impulse is 0.6 Ns .

(a) Find the greatest possible speed of $P$ after the impulse acts.
(b) Find the least possible speed of $P$ after the impulse acts.
In fact the speed of $P$ after the impulse acts is $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the angle the impulse makes with the original direction of travel of $P$ and draw a sketch to make this direction clear.

OCR M3 2014 June Q2

2 One end of a light elastic string, of natural length 0.6 m and modulus of elasticity 30 N , is attached to a fixed point $O$. A particle $P$ of weight 48 N is attached to the other end of the string. $P$ is released from rest at a point $d \mathrm {~m}$ vertically below $O$. Subsequently $P$ just reaches $O$.

Find $d$.
Find the magnitude and direction of the acceleration of $P$ when it has travelled 1.3 m from its point of release.

OCR M3 2014 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-2_403_951_1247_559} 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. \section*{$\mathrm { OCR } ^ { \text {勾 } }$}

OCR M3 2014 June Q4

4 A particle $P$ of mass 0.4 kg is projected horizontally with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a fixed point $O$ on a smooth horizontal surface. At time $t \mathrm {~s}$ after projection $P$ is $x \mathrm {~m}$ from $O$ and is moving away from $O$ with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. There is a force of magnitude $1.6 v ^ { 2 } \mathrm {~N}$ resisting the motion of $P$.

Find an expression for $\frac { \mathrm { d } v } { \mathrm {~d} x }$ in terms of $v$, and hence show that $v = 2 \mathrm { e } ^ { - 4 x }$.
Find the distance travelled by $P$ in the 0.5 seconds after it leaves $O$.

OCR M3 2014 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-3_510_716_662_676} Two uniform rods $A B$ and $B C$, each of length $4 L$, are freely jointed at $B$, and rest in a vertical plane with $A$ and $C$ on a smooth horizontal surface. The weight of $A B$ is $W$ and the weight of $B C$ is $2 W$. The rods are joined by a horizontal light inextensible string fixed to each rod at a point distance $L$ from $B$, so that each rod is inclined at an angle of $60 ^ { \circ }$ to the horizontal (see diagram).

By considering the equilibrium of the whole body, show that the force acting on $B C$ at $C$ is $1.75 W$ and find the force acting on $A B$ at $A$.
Find the tension in the string in terms of $W$.
Find the horizontal and vertical components of the force acting on $A B$ at $B$, and state the direction of the component in each case.

OCR M3 2014 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_547_515_267_772} A hollow cylinder is fixed with its axis horizontal. $O$ is the centre of a vertical cross-section of the cylinder and $D$ is the highest point on the cross-section. $A$ and $C$ are points on the circumference of the cross-section such that $A O$ and $C O$ are both inclined at an angle of $30 ^ { \circ }$ below the horizontal diameter through $O$. The inner surface of the cylinder is smooth and has radius 0.8 m (see diagram). A particle $P$, of mass $m \mathrm {~kg}$, and a particle $Q$, of mass $5 m \mathrm {~kg}$, are simultaneously released from rest from $A$ and $C$, respectively, inside the cylinder. $P$ and $Q$ collide; the coefficient of restitution between them is 0.95 .

Show that, immediately after the collision, $P$ moves with speed $6.3 \mathrm {~ms} ^ { - 1 }$, and find the speed and direction of motion of $Q$.
Find, in terms of $m$, an expression for the normal reaction acting on $P$ when it subsequently passes through $D$.

OCR M3 2014 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_382_773_1567_648} One end of a light elastic string, of natural length 0.3 m , is attached to a fixed point $O$ on a smooth plane that is inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = 0.2$. A particle $P$ of mass $m \mathrm {~kg}$ is attached to the other end of the string. The string lies along a line of greatest slope of the plane and has modulus of elasticity $2.45 m \mathrm {~N}$ (see diagram).

Show that in the equilibrium position the extension of the string is 0.24 m .
$P$ is given a velocity of $0.3 \mathrm {~ms} ^ { - 1 }$ down the plane from the equilibrium position.
Show that $P$ performs simple harmonic motion with period 2.20 s (correct to 3 significant figures), and find the amplitude of the motion.
Find the distance of $P$ from $O$ and the velocity of $P$ at the instant 1.5 seconds after $P$ is set in motion.

OCR M3 2015 June Q1

1 A particle $P$ of mass 0.2 kg is moving on a smooth horizontal surface with speed $3 \mathrm {~ms} ^ { - 1 }$, when it is struck by an impulse of magnitude $I$ Ns. The impulse acts horizontally in a direction perpendicular to the original direction of motion of $P$, and causes the direction of motion of $P$ to change by an angle $\alpha$, where $\tan \alpha = \frac { 5 } { 12 }$.

Show that $I = 0.25$.
Find the speed of $P$ after the impulse acts.

OCR M3 2015 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{2734e846-f640-4203-ac11-6b2180a21950-2_556_736_671_667} Two uniform rods $A B$ and $B C$, each of length $2 L$, are freely jointed at $B$, and $A B$ is freely jointed to a fixed point at $A$. The rods are held in equilibrium in a vertical plane by a light horizontal string attached at $C$. The rods $A B$ and $B C$ make angles $\alpha$ and $\beta$ to the horizontal respectively. The weight of $\operatorname { rod } B C$ is 75 N , and the tension in the string is 50 N (see diagram).

Show that $\tan \beta = \frac { 3 } { 4 }$.
Given that $\tan \alpha = \frac { 12 } { 5 }$, find the weight of $A B$.

OCR M3 2015 June Q4

4 A particle of mass 0.4 kg , moving on a smooth horizontal surface, passes through a point $O$ with velocity $10 \mathrm {~ms} ^ { - 1 }$. At time $t \mathrm {~s}$ after the particle passes through $O$, the particle has a displacement $x \mathrm {~m}$ from $O$, has a velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ away from $O$, and is acted on by a force of magnitude $\frac { 1 } { 8 } v \mathrm {~N}$ acting towards $O$. Find

the time taken for the velocity of the particle to reduce from $10 \mathrm {~ms} ^ { - 1 }$ to $5 \mathrm {~ms} ^ { - 1 }$,
the average velocity of the particle over this time.

OCR M3 2015 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{2734e846-f640-4203-ac11-6b2180a21950-4_337_944_255_557} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses $2 m \mathrm {~kg}$ and $m \mathrm {~kg}$ respectively. The spheres are moving on a horizontal surface when they collide. Before the collision, $A$ is moving with speed $a \mathrm {~ms} ^ { - 1 }$ in a direction making an angle $\alpha$ with the line of centres and $B$ is moving towards $A$ with speed $b \mathrm {~ms} ^ { - 1 }$ in a direction making an angle $\beta$ with the line of centres (see diagram). After the collision, $A$ moves with velocity $2 \mathrm {~ms} ^ { - 1 }$ in a direction perpendicular to the line of centres and $B$ moves with velocity $2 \mathrm {~ms} ^ { - 1 }$ in a direction making an angle of $45 ^ { \circ }$ with the line of centres. The coefficient of restitution between $A$ and $B$ is $\frac { 2 } { 3 }$.

Show that $a \cos \alpha = \frac { 5 } { 6 } \sqrt { 2 }$ and find $b \cos \beta$.
Find the values of $a$ and $\alpha$.

OCR M3 2015 June Q6

6 A particle $P$ starts from rest from a point $A$ and moves in a straight line with simple harmonic motion about a point $O$. At time $t$ seconds after the motion starts the displacement of $P$ from $O$ is $x \mathrm {~m}$ towards $A$. The particle $P$ is next at rest when $t = 0.25 \pi$ having travelled a distance of 1.2 m .

Find the maximum velocity of $P$.
Find the value of $x$ and the velocity of $P$ when $t = 0.7$.
Find the other values of $t$, for $0 < t < 1$, at which $P$ 's speed is the same as when $t = 0.7$. Find also the corresponding values of $x$.

OCR M3 2015 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{2734e846-f640-4203-ac11-6b2180a21950-4_282_474_1809_794} One end of a light inextensible string of length 0.5 m is attached to a fixed point $O$. A particle $P$ of mass 0.2 kg is attached to the other end of the string. $P$ is projected horizontally from the point 0.5 m below $O$ with speed $u \mathrm {~ms} ^ { - 1 }$. When the string makes an angle of $\theta$ with the downward vertical the particle has speed $v \mathrm {~ms} ^ { - 1 }$ (see diagram).

Show that, while the string is taut, the tension, $T \mathrm {~N}$, in the string is given by $$T = 5.88 \cos \theta + 0.4 u ^ { 2 } - 3.92 .$$
Find the least value of $u$ for which the particle will move in a complete circle.
If in fact $u = 3.5 \mathrm {~ms} ^ { - 1 }$, find the speed of the particle at the point where the string first becomes slack.

OCR M3 2016 June Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-2_285_1096_255_488} A particle $P$ of mass 0.3 kg is moving with speed $0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a straight line on a smooth horizontal surface when it is struck by a horizontal impulse. After the impulse acts $P$ has speed $0.6 \mathrm {~ms} ^ { - 1 }$ and is moving in a direction making an angle $30 ^ { \circ }$ with its original direction of motion (see diagram).

Find the magnitude of the impulse and the angle its line of action makes with the original direction of motion of $P$. Subsequently a second impulse acts on $P$. After this second impulse acts, $P$ again moves from left to right with speed $0.4 \mathrm {~ms} ^ { - 1 }$ in a direction parallel to its original direction of motion.
State the magnitude of the second impulse, and show the direction of the second impulse on a diagram.

OCR M3 2016 June Q2

2 A particle $Q$ of mass 0.2 kg is projected horizontally with velocity $4 \mathrm {~ms} ^ { - 1 }$ from a fixed point $A$ on a smooth horizontal surface. At time $t \mathrm {~s}$ after projection $Q$ is $x \mathrm {~m}$ from $A$ and is moving away from $A$ with velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. There is a force of $3 \cos 2 t \mathrm {~N}$ acting on $Q$ in the positive $x$-direction.

Find an expression for the velocity of $Q$ at time $t$. State the maximum and minimum values of the velocity of $Q$ as $t$ varies.
Find the average velocity of $Q$ between times $t = \pi$ and $t = \frac { 3 } { 2 } \pi$.
\includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-2_549_1237_1724_415} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses $2 m \mathrm {~kg}$ and $3 m \mathrm {~kg}$ respectively. The spheres are approaching each other on a horizontal surface when they collide. Before the collision $A$ is moving with speed $5 \mathrm {~ms} ^ { - 1 }$ in a direction making an angle $\alpha$ with the line of centres, where $\cos \alpha = \frac { 4 } { 5 }$, and $B$ is moving with speed $3 \frac { 1 } { 4 } \mathrm {~ms} ^ { - 1 }$ in a direction making an angle $\beta$ with the line of centres, where $\cos \beta = \frac { 5 } { 13 }$. A straight vertical wall is situated to the right of $B$, perpendicular to the line of centres (see diagram). The coefficient of restitution between $A$ and $B$ is $\frac { 2 } { 3 }$.

OCR M3 2016 June Q7

7 A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of modulus of elasticity 24 mgN and natural length 0.6 m . The other end of the string is attached to a fixed point $O$; the particle $P$ rests in equilibrium at a point $A$ with the string vertical.

Show that the distance $O A$ is 0.625 m . Another particle $Q$, of mass $3 m \mathrm {~kg}$, is released from rest from a point 0.4 m above $P$ and falls onto $P$. The two particles coalesce.
Show that the combined particle initially moves with speed $2.1 \mathrm {~ms} ^ { - 1 }$.
Show that the combined particle initially performs simple harmonic motion, and find the centre of this motion and its amplitude.
Find the time that elapses between $Q$ being released from rest and the combined particle first reaching the highest point of its subsequent motion. \section*{END OF QUESTION PAPER}

OCR M3 Specimen Q1

1 A particle is moving with simple harmonic motion in a straight line. The period is 0.2 s and the amplitude of the motion is 0.3 m . Find the maximum speed and the maximum acceleration of the particle.

OCR M3 Specimen Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-2_296_798_461_694} A sphere $A$ of mass $m$, moving on a horizontal surface, collides with another sphere $B$ of mass $2 m$, which is at rest on the surface. The spheres are smooth and uniform, and have equal radius. Immediately before the collision, $A$ has velocity $u$ at an angle $\theta ^ { \circ }$ to the line of centres of the spheres (see diagram). Immediately after the collision, the spheres move in directions that are perpendicular to each other.

Find the coefficient of restitution between the spheres.
Given that the spheres have equal speeds after the collision, find $\theta$.

OCR M3 Specimen Q3

3 An aircraft of mass 80000 kg travelling at $90 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ touches down on a straight horizontal runway. It is brought to rest by braking and resistive forces which together are modelled by a horizontal force of magnitude ( $27000 + 50 v ^ { 2 }$ ) newtons, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the aircraft. Find the distance travelled by the aircraft between touching down and coming to rest.

OCR M3 Specimen Q4

4 For a bungee jump, a girl is joined to a fixed point $O$ of a bridge by an elastic rope of natural length 25 m and modulus of elasticity 1320 N . The girl starts from rest at $O$ and falls vertically. The lowest point reached by the girl is 60 m vertically below $O$. The girl is modelled as a particle, the rope is assumed to be light, and air resistance is neglected.

Find the greatest tension in the rope during the girl's jump.
Use energy considerations to find
(a) the mass of the girl,
(b) the speed of the girl when she has fallen half way to the lowest point.

OCR M3 Specimen Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_576_535_258_804} A particle $P$ of mass 0.3 kg is moving in a vertical circle. It is attached to the fixed point $O$ at the centre of the circle by a light inextensible string of length 1.5 m . When the string makes an angle of $40 ^ { \circ }$ with the downward vertical, the speed of $P$ is $6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see diagram). Air resistance may be neglected.

Find the radial and transverse components of the acceleration of $P$ at this instant. In the subsequent motion, with the string still taut and making an angle $\theta ^ { \circ }$ with the downward vertical, the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Use conservation of energy to show that $v ^ { 2 } \approx 19.7 + 29.4 \cos \theta ^ { \circ }$.
Find the tension in the string in terms of $\theta$.
Find the value of $v$ at the instant when the string becomes slack.
\includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_574_842_1640_664} A step-ladder is modelled as two uniform rods $A B$ and $A C$, freely jointed at $A$. The rods are in equilibrium in a vertical plane with $B$ and $C$ in contact with a rough horizontal surface. The rods have equal lengths; $A B$ has weight 150 N and $A C$ has weight 270 N . The point $A$ is 2.5 m vertically above the surface, and $B C = 1.6 \mathrm {~m}$ (see diagram).
Find the horizontal and vertical components of the force acting on $A C$ at $A$.
The coefficient of friction has the same value $\mu$ at $B$ and at $C$, and the step-ladder is on the point of slipping. Giving a reason, state whether the equilibrium is limiting at $B$ or at $C$, and find $\mu$.
\includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-4_648_227_269_982} Two points $A$ and $B$ lie on a vertical line with $A$ at a distance 2.6 m above $B$. A particle $P$ of mass 10 kg is joined to $A$ by an elastic string and to $B$ by another elastic string (see diagram). Each string has natural length 0.8 m and modulus of elasticity 196 N . The strings are light and air resistance may be neglected.
Verify that $P$ is in equilibrium when $P$ is vertically below $A$ and the length of the string $P A$ is 1.5 m . The particle is set in motion along the line $A B$ with both strings remaining taut. The displacement of $P$ below the equilibrium position is denoted by $x$ metres.
Show that the tension in the string $P A$ is $245 ( 0.7 + x )$ newtons, and the tension in the string $P B$ is $245 ( 0.3 - x )$ newtons.
Show that the motion of $P$ is simple harmonic.
Given that the amplitude of the motion is 0.25 m , find the proportion of time for which $P$ is above the mid-point of $A B$.

OCR M3 2009 January Q4

Show that $v ^ { 2 } = 9 + 9.8 \sin \theta$.
Find, in terms of $\theta$, the radial and tangential components of the acceleration of $P$.
Show that the tension in the string is $( 3.6 + 5.88 \sin \theta ) \mathrm { N }$ and hence find the value of $\theta$ at the instant when the string becomes slack, giving your answer correct to 1 decimal place.

Questions — OCR M3 (130 questions)

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