Questions - OCR M3 - A-Level Maths

Show that the magnitude of the frictional force at $C$ is 27 N .
Calculate the horizontal and vertical components of the force exerted on $A B$ at $B$.
Given that friction is limiting at $C$, find the coefficient of friction between the $\operatorname { rod } B C$ and the wall.

OCR M3 Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_598_839_1480_706} One end of a light inextensible string of length 0.5 m is attached to a fixed point $O$. A particle $P$ of mass 0.3 kg is attached to the other end of the string. With the string taut and at an angle of $60 ^ { \circ }$ to the upward vertical, $P$ is projected with speed $2 \mathrm {~ms} ^ { - 1 }$ (see diagram). $P$ begins to move without air resistance in a vertical circle with centre $O$. When the string makes an angle $\theta$ with the upward vertical, the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $v ^ { 2 } = 8.9 - 9.8 \cos \theta$.
Find the tension in the string in terms of $\theta$.
$P$ does not move in a complete circle. Calculate the angle through which $O P$ turns before $P$ leaves the circular path.

OCR M3 2006 January Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_246_693_278_731} A particle $P$ of mass 0.4 kg moving in a straight line has speed $8.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. An impulse applied to $P$ deflects it through $45 ^ { \circ }$ and reduces its speed to $5.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see diagram). Calculate the magnitude and direction of the impulse exerted on $P$.
$2 \quad O$ is a fixed point on a horizontal straight line. A particle $P$ of mass 0.5 kg is released from rest at $O$. At time $t$ seconds after release the only force acting on $P$ has magnitude $\left( 1 + k t ^ { 2 } \right) \mathrm { N }$ and acts horizontally and away from $O$ along the line, where $k$ is a positive constant.

Find the speed of $P$ in terms of $k$ and $t$.
Given that $P$ is 2 m from $O$ when $t = 1$, find the value of $k$ and the time taken by $P$ to travel 20 m from $O$.

OCR M3 2006 January Q3

3 A light elastic string has natural length 3 m . One end is attached to a fixed point $O$ and the other end is attached to a particle of mass 1.6 kg . The particle is released from rest in a position 5 m vertically below $O$. Air resistance may be neglected.

Given that in the subsequent motion the particle just reaches $O$, show that the modulus of elasticity of the string is 117.6 N .
Calculate the speed of the particle when it is 4.5 m below $O$.

OCR M3 2006 January Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_283_711_1754_722} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, $A$ has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is moving perpendicular to the line of centres, and $B$ has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

OCR M3 2006 January Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_462_1109_283_569} Two uniform rods $A B$ and $B C$ have weights 64 N and 40 N respectively. The rods are freely jointed to each other at $B$. The rod $A B$ is freely jointed to a fixed point on horizontal ground at $A$ and the rod $B C$ rests against a vertical wall at $C$. The rod $B C$ is 1.8 m long and is horizontal. A particle of weight 9 N is attached to the rod $B C$ at the point 0.4 m from $C$. The point $A$ is 1.2 m below the level of $B C$ and 3.8 m from the wall (see diagram). The system is in equilibrium.

Show that the magnitude of the frictional force at $C$ is 27 N .
Calculate the horizontal and vertical components of the force exerted on $A B$ at $B$.
Given that friction is limiting at $C$, find the coefficient of friction between the $\operatorname { rod } B C$ and the wall.

OCR M3 2006 January Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_598_839_1480_706} One end of a light inextensible string of length 0.5 m is attached to a fixed point $O$. A particle $P$ of mass 0.3 kg is attached to the other end of the string. With the string taut and at an angle of $60 ^ { \circ }$ to the upward vertical, $P$ is projected with speed $2 \mathrm {~ms} ^ { - 1 }$ (see diagram). $P$ begins to move without air resistance in a vertical circle with centre $O$. When the string makes an angle $\theta$ with the upward vertical, the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $v ^ { 2 } = 8.9 - 9.8 \cos \theta$.
Find the tension in the string in terms of $\theta$.
$P$ does not move in a complete circle. Calculate the angle through which $O P$ turns before $P$ leaves the circular path.

OCR M3 2006 January Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-4_122_1009_265_571} As shown in the diagram, $A$ and $B$ are fixed points on a smooth horizontal table, where $A B = 3 \mathrm {~m}$. A particle $Q$ of mass 1.2 kg is attached to $A$ by a light elastic string of natural length 1 m and modulus of elasticity $180 \mathrm {~N} . Q$ is attached to $B$ by a light elastic string of natural length 1.2 m and modulus of elasticity 360 N .

Verify that when $Q$ is in equilibrium $B Q = 1.5 \mathrm {~m}$.
$Q$ is projected towards $B$ from the equilibrium position with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Subsequently $Q$ oscillates with simple harmonic motion.
Show that the period of the motion is 0.314 s approximately.
Show that $u \leqslant 6$.
Given that $u = 6$, find the time taken for $Q$ to move from the equilibrium position to a position 1.3 m from $A$ for the first time.

OCR M3 2007 January Q1

1 A particle $P$ of mass 0.6 kg is attached to a fixed point $O$ by a light inextensible string of length 0.4 m . While hanging at a distance 0.4 m vertically below $O , P$ is projected horizontally with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and moves in a complete vertical circle. Calculate the tension in the string when $P$ is vertically above $O$.

OCR M3 2007 January Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-2_231_971_539_587} When a tennis ball of mass 0.057 kg bounces it receives an impulse of magnitude $I \mathrm {~N} \mathrm {~s}$ at an angle of $\theta$ to the horizontal. Immediately before the ball bounces it has speed $28 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction of $30 ^ { \circ }$ to the horizontal. Immediately after the ball bounces it has speed $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction of $30 ^ { \circ }$ to the horizontal (see diagram). Find $I$ and $\theta$.

OCR M3 2007 January Q3

3 marks

3
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-2_465_757_1146_694} Two identical uniform rods, $A B$ and $B C$, are freely jointed to each other at $B$, and $A$ is freely jointed to a fixed point. The rods are in limiting equilibrium in a vertical plane, with $C$ resting on a rough horizontal surface. $A B$ is horizontal, and $B C$ is inclined at $60 ^ { \circ }$ to the horizontal. The weight of each rod is 160 N (see diagram).

By taking moments for $A B$ about $A$, find the vertical component of the force on $A B$ at $B$. Hence or otherwise find the magnitude of the vertical component of the contact force on $B C$ at $C$. [3]
Calculate the magnitude of the frictional force on $B C$ at $C$ and state its direction.
Calculate the value of the coefficient of friction at $C$.

OCR M3 2007 January Q4

4 A particle $P$ of mass 0.2 kg is suspended from a fixed point $O$ by a light elastic string of natural length 0.7 m and modulus of elasticity $3.5 \mathrm {~N} . P$ is at the equilibrium position when it is projected vertically downwards with speed $1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t \mathrm {~s}$ after being set in motion $P$ is $x \mathrm {~m}$ below the equilibrium position and has velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that the equilibrium position of $P$ is 1.092 m below $O$.
Prove that $P$ moves with simple harmonic motion, and calculate the amplitude.
Calculate $x$ and $v$ when $t = 0.4$.

OCR M3 2007 January Q5

5 The pilot of a hot air balloon keeps it at a fixed altitude by dropping sand from the balloon. Each grain of sand has mass $m \mathrm {~kg}$ and is released from rest. When a grain has fallen a distance $x \mathrm {~m}$, it has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Each grain falls vertically and the only forces acting on it are its weight and air resistance of magnitude $m k v ^ { 2 } \mathrm {~N}$, where $k$ is a positive constant.

Show that $\left( \frac { v } { g - k v ^ { 2 } } \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1$.
Find $v ^ { 2 }$ in terms of $k , g$ and $x$. Hence show that, as $x$ becomes large, the limiting value of $v$ is $\sqrt { \frac { g } { k } }$.
Given that the altitude of the balloon is 300 m and that each grain strikes the ground at $90 \%$ of its limiting velocity, find $k$.

OCR M3 2007 January Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-3_446_821_1007_664} Two uniform smooth spheres $A$ and $B$ of equal radius are moving on a horizontal surface when they collide. $A$ has mass 0.4 kg , and $B$ has mass $m \mathrm {~kg}$. Immediately before the collision, $A$ is moving with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an acute angle $\theta$ to the line of centres, and $B$ is moving with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $30 ^ { \circ }$ to the line of centres. Immediately after the collision $A$ is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $45 ^ { \circ }$ to the line of centres, and $B$ is moving with speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ perpendicular to the line of centres (see diagram).

Find $u$.
Given that $\theta = 88.1 ^ { \circ }$ correct to 1 decimal place, calculate the approximate values of $v$ and $m$.
The coefficient of restitution is 0.75 . Show that the exact value of $\theta$ is a root of the equation $8 \sin \theta - 6 \cos \theta = 9 \cos 30 ^ { \circ }$.

OCR M3 2007 January Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-4_721_691_269_726} The diagram shows a particle $P$ of mass 0.5 kg attached to the highest point $A$ of a fixed smooth sphere by a light elastic string. The sphere has centre $O$ and radius 1.2 m . The string has natural length 0.6 m and modulus of elasticity $6.86 \mathrm {~N} . P$ is released from rest at a point on the surface of the sphere where the acute angle $A O P$ is at least 0.5 radians.

(a) For the case angle $A O P = \alpha , P$ remains at rest. Show that $\sin \alpha = 2.8 \alpha - 1.4$.
(b) Use the iterative formula $$\alpha _ { n + 1 } = \frac { \sin \alpha _ { n } } { 2.8 } + 0.5 ,$$ with $\alpha _ { 1 } = 0.8$, to find $\alpha$ correct to 2 significant figures.
Given instead that angle $A O P = 0.5$ radians when $P$ is released, find the speed of $P$ when angle $A O P = 0.8$ radians, given that $P$ is at all times in contact with the surface of the sphere. State whether the speed of $P$ is increasing or decreasing when angle $A O P = 0.8$ radians.

OCR M3 2008 January Q1

1 A smooth horizontal surface lies in the $x - y$ plane. A particle $P$ of mass 0.5 kg is moving on the surface with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the $x$-direction when it is struck by a horizontal blow whose impulse has components - 3.5 N s and 2.4 N s in the $x$-direction and $y$-direction respectively.

Find the components in the $x$-direction and the $y$-direction of the velocity of $P$ immediately after the blow. Hence show that the speed of $P$ immediately after the blow is $5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
$P$ is struck by a second horizontal blow whose impulse is $\mathbf { I }$.
Given that $P$ 's direction of motion immediately after this blow is parallel to the $x$-axis, write down the component of $\mathbf { I }$ in the $y$-direction.

OCR M3 2008 January Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-2_515_1065_861_541} Two uniform rods $A B$ and $B C$, each of length 2 m , are freely jointed at $B$. The weights of the rods are $W \mathrm {~N}$ and 50 N respectively. The end $A$ of $A B$ is hinged at a fixed point. The rods $A B$ and $B C$ make angles $\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)$ and $\beta$ respectively with the downward vertical, and are held in equilibrium in a vertical plane by a horizontal force of magnitude 75 N acting at $C$ (see diagram).

By taking moments about $B$ for $B C$, show that $\tan \beta = 3$.
Write down the horizontal and vertical components of the force acting on $A B$ at $B$.
Find the value of $W$.

OCR M3 2008 January Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-3_419_921_267_612} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses 6 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision the velocity of $A$ has components $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the line of centres towards $B$, and $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ perpendicular to the line of centres. $B$ is moving with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the line of centres towards $A$ (see diagram). The coefficient of restitution between the spheres is $e$.

Find, in terms of $e$, the component of the velocity of $A$ along the line of centres immediately after the collision.
Given that the speeds of $A$ and $B$ are the same immediately after the collision, and that $3 e ^ { 2 } = 1$, find $v$.

OCR M3 2008 January Q4

4 A particle of mass $m \mathrm {~kg}$ is released from rest at a fixed point $O$ and falls vertically. The particle is subject to an upward resisting force of magnitude $0.49 m v \mathrm {~N}$ where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity of the particle when it has fallen a distance of $x \mathrm {~m}$ from $O$.

Write down a differential equation for the motion of the particle, and show that the equation can be written as $\left( \frac { 20 } { 20 - v } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 0.49$.
Hence find an expression for $x$ in terms of $v$.

OCR M3 2008 January Q5

5 A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 0.75 mgN . The other end of the string is attached to a fixed point $O$ of a smooth plane inclined at $30 ^ { \circ }$ to the horizontal. $P$ is released from rest at $O$ and moves down the plane.

Show that the maximum speed of $P$ is reached when the extension of the string is 0.8 m .
Find the maximum speed of $P$.
Find the maximum displacement of $P$ from $O$.

OCR M3 2008 January Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-4_497_524_276_804} A particle $P$ of mass 0.4 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point $O$. With the string taut the particle is travelling in a circular path in a vertical plane. The angle between the string and the downward vertical is $\theta ^ { \circ }$ (see diagram). When $\theta = 0$ the speed of $P$ is $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

At the instant when the string is horizontal, find the speed of $P$ and the tension in the string.
At the instant when the string becomes slack, find the value of $\theta$.

OCR M3 2008 January Q7

7 A particle $P$, of mass $m \mathrm {~kg}$, is attached to one end of a light elastic string of natural length 3.2 m and modulus of elasticity $4 m g \mathrm {~N}$. The other end of the string is attached to a fixed point $A$. The particle is released from rest at a point 4.8 m vertically below $A$. At time $t \mathrm {~s}$ after $P$ 's release $P$ is ( $4 + x ) \mathrm { m }$ below $A$.

Show that $4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 49 x$.
$P$ 's motion is simple harmonic.
Write down the amplitude of $P$ 's motion and show that the string becomes slack instantaneously at intervals of approximately 1.8 s . A particle $Q$ is attached to one end of a light inextensible string of length $L \mathrm {~m}$. The other end of the string is attached to a fixed point $B$. The particle is released from rest with the string taut and inclined at a small angle with the downward vertical. At time $t \mathrm {~s}$ after $Q$ 's release $B Q$ makes an angle of $\theta$ radians with the downward vertical.
Show that $\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } \approx - \frac { g } { L } \theta$. The period of the simple harmonic motion to which $Q$ 's motion approximates is the same as the period of $P$ 's motion.
Given that $\theta = 0.08$ when $t = 0$, find the speed of $Q$ when $t = 0.25$.

OCR M3 2009 January Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-2_385_741_269_701} A particle $P$ of mass 0.5 kg is moving in a straight line with speed $6.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. An impulse of magnitude 2.6 N s applied to $P$ deflects its direction of motion through an angle $\theta$, and reduces its speed to $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see diagram). By considering an impulse-momentum triangle, or otherwise,

show that $\cos \theta = 0.6$,
find the angle that the impulse makes with the original direction of motion of $P$.

OCR M3 2009 January Q2

2 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{14403602-94a6-4441-a673-65f9b98180e5-2_501_752_1133_356} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{14403602-94a6-4441-a673-65f9b98180e5-2_519_558_1183_1231} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Two uniform rods $A B$ and $B C$, of weights 70 N and 110 N respectively, are freely jointed at $B$. The rods are in equilibrium in a vertical plane with $A$ and $C$ at the same horizontal level and $A C = 2 \mathrm {~m}$. The $\operatorname { rod } A B$ is freely jointed to a fixed point at $A$ and the rod $B C$ is freely jointed to a fixed point at $C$. The horizontal distance between $B$ and $A$ is 4 m and $B$ is 4 m below $A C$; angle $B A C$ is obtuse (see Fig. 1). The force exerted on the $\operatorname { rod } A B$ at $B$, by the $\operatorname { rod } B C$, has horizontal and vertical components as shown in Fig. 2.

By taking moments about $A$ for the $\operatorname { rod } A B$ find the value of $X - Y$.
By taking moments about $C$ for the rod $B C$ show that $2 X - 3 Y + 165 = 0$.
Find the magnitude of the force acting between $A B$ and $B C$ at $B$.

Questions — OCR M3 (130 questions)

Browse by module

Browse by board

OCR M3 Q4

OCR M3 Q5

OCR M3 Q6

OCR M3 2006 January Q1

OCR M3 2006 January Q3

OCR M3 2006 January Q4

OCR M3 2006 January Q5

OCR M3 2006 January Q6

OCR M3 2006 January Q7

OCR M3 2007 January Q1

OCR M3 2007 January Q2

OCR M3 2007 January Q3

OCR M3 2007 January Q4

OCR M3 2007 January Q5

OCR M3 2007 January Q6

OCR M3 2007 January Q7

OCR M3 2008 January Q1

OCR M3 2008 January Q2

OCR M3 2008 January Q3

OCR M3 2008 January Q4

OCR M3 2008 January Q5

OCR M3 2008 January Q6

OCR M3 2008 January Q7

OCR M3 2009 January Q1

OCR M3 2009 January Q2