Questions - CAIE M2 - A-Level Maths

Show that the distance between the centre of mass of the sphere and the centre of the sphere is 0.005 m . This sphere is placed on a horizontal surface with its axis of symmetry horizontal. The equilibrium of the sphere is maintained by a force of magnitude $F \mathrm {~N}$ acting parallel to the axis of symmetry applied to the highest point of the sphere.
Calculate $F$.

CAIE M2 2012 June Q3

3 A light elastic string has natural length 2.2 m and modulus of elasticity 14.3 N . A particle $P$ of mass $m \mathrm {~kg}$ is attached to the mid-point of the string. The ends of the string are attached to fixed points $A$ and $B$ which are 2.4 m apart at the same horizontal level. $P$ is released from rest at the mid-point of $A B$. In the subsequent motion $P$ has its greatest speed at a point 0.5 m below $A B$.

Find $m$.
Calculate the greatest speed of $P$.

CAIE M2 2012 June Q4

4 A particle $P$ of mass 0.25 kg moves in a straight line on a smooth horizontal surface. At time $t \mathrm {~s}$ the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A variable force of magnitude $3 t \mathrm {~N}$ opposes the motion of $P$.

Given that $P$ comes to rest when $t = 3$, find $v$ when $t = 0$.
Calculate the distance travelled by $P$ in the interval $0 \leqslant t \leqslant 3$.

CAIE M2 2012 June Q5

5 A ball is projected with velocity $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $70 ^ { \circ }$ above the horizontal from a point $O$ on horizontal ground. The ball subsequently bounces once on the ground at a point $P$ before landing at a point $Q$ where it remains at rest. The distance $P Q$ is 17.1 m .

Calculate the time taken by the ball to travel from $O$ to $P$ and the distance $O P$.
Given that the horizontal component of the velocity of the ball does not change at $P$, calculate the speed of the ball when it leaves $P$.

CAIE M2 2012 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{09971be0-73b6-4c73-8dfd-c89ff877950a-3_451_775_255_685} The diagram shows a uniform lamina $A B C D E F$, formed from a semicircle with centre $O$ and radius 1 m by removing a semicircular part with centre $O$ and radius $r \mathrm {~m}$.

Show that the distance in metres of the centre of mass of the lamina from $O$ is $$\frac { 4 \left( 1 + r + r ^ { 2 } \right) } { 3 \pi ( 1 + r ) } .$$ The centre of mass of the lamina lies on the $\operatorname { arc } A B C$.
Show that $r = 0.494$, correct to 3 significant figures. The lamina is freely suspended at $F$ and hangs in equilibrium.
Find the angle between the diameter of the lamina and the vertical.

CAIE M2 2012 June Q7

6 marks

7 Particles $P$ and $Q$, of masses 0.8 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string which passes through a small hole in a smooth horizontal table of negligible thickness. $P$ moves with constant angular speed $6.25 \mathrm { rad } \mathrm { s } ^ { - 1 }$ in a circular path on the surface of the table.

It is given that $Q$ is stationary and that the part of string attached to $Q$ is vertical. Calculate the radius of the path of $P$, and find the speed of $P$.
It is given instead that the part of string attached to $Q$ is inclined at $60 ^ { \circ }$ to the vertical, and that $Q$ moves in a horizontal circular path below the table, also with constant angular speed $6.25 \mathrm { rad } \mathrm { s } ^ { - 1 }$. Calculate the total length of the string.
[0pt] [6]

CAIE M2 2012 June Q1

1 A particle $P$ is projected with speed $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ above the horizontal from a point $O$ on horizontal ground. Calculate the distance $O P$ at the instant 2 s after projection.

CAIE M2 2012 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_448_547_434_799} The diagram shows a uniform object $A B C$ of weight 3 N in the form of an arc of a circle with centre $O$ and radius 0.7 m . The angle $A O C$ is 2 radians. The object rests in equilibrium with $A$ on a horizontal surface and $C$ vertically above $A$. Equilibrium is maintained by a horizontal force of magnitude $F \mathrm {~N}$ applied at $C$ in the plane of the object. Calculate $F$.

CAIE M2 2012 June Q3

3 A particle $P$ of mass 0.2 kg is projected horizontally from a fixed point $O$, and moves in a straight line on a smooth horizontal surface. A force of magnitude $0.4 x \mathrm {~N}$ acts on $P$ in the direction $P O$, where $x \mathrm {~m}$ is the displacement of $P$ from $O$.

Given that $P$ comes to instantaneous rest when $x = 2.5$, find the initial kinetic energy of $P$.
Find the value of $x$ on the first occasion when the speed of $P$ is $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

CAIE M2 2012 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_170_616_1649_767} A small sphere $S$ of mass $m \mathrm {~kg}$ is moving inside a fixed smooth hollow cylinder whose axis is vertical. $S$ moves with constant speed in a horizontal circle of radius 0.4 m and is in contact with both the plane base and the curved surface of the cylinder (see diagram).

Given that the horizontal and vertical forces exerted on $S$ by the cylinder have equal magnitudes, calculate the speed of $S$.
$S$ is now attached to the centre of the base of the cylinder by a horizontal light elastic string of natural length 0.25 m and modulus of elasticity 13 N . The sphere $S$ is set in motion and moves in a horizontal circle with constant angular speed $\omega \mathrm { rads } ^ { - 1 }$ and is in contact with both the plane base and the curved surface of the cylinder.
It is given that the magnitudes of the horizontal and vertical forces exerted on $S$ by the cylinder are equal if $\omega = 8$. Calculate $m$.
For the value of $m$ found in part (ii), find the least possible value of $\omega$ for the motion.

CAIE M2 2012 June Q5

5 A light elastic string has natural length 3 m and modulus of elasticity 45 N . A particle $P$ of mass 0.6 kg is attached to the mid-point of the string. The ends of the string are attached to fixed points $A$ and $B$ which lie on a line of greatest slope of a smooth plane inclined at $30 ^ { \circ }$ to the horizontal. The distance $A B$ is 4 m , and $A$ is higher than $B$.

Calculate the distance $A P$ when $P$ rests on the slope in equilibrium.
$P$ is released from rest at the point between $A$ and $B$ where $A P = 2.5 \mathrm {~m}$.
Find the maximum speed of $P$.
Show that $P$ is at rest when $A P = 1.6 \mathrm {~m}$.

CAIE M2 2012 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-3_341_791_886_678} A uniform lamina $A B C D E$ consists of a rectangle $B C D E$ and an isosceles triangle $A B E$ joined along their common edge $B E$. For the triangle, $A B = A E , B E = a \mathrm {~m}$ and the perpendicular height is $h \mathrm {~m}$. For the rectangle, $B C = D E = 0.5 \mathrm {~m}$ and $C D = B E = a \mathrm {~m}$ (see diagram).

Show that the distance in metres of the centre of mass of the lamina from $B E$ towards $C D$ is $$\frac { 3 - 4 h ^ { 2 } } { 12 + 12 h }$$ The lamina is freely suspended at $E$ and hangs in equilibrium.
Given that $D E$ is horizontal, calculate $h$.
Given instead that $h = 0.5$ and $A E$ is horizontal, calculate $a$.

CAIE M2 2012 June Q7

7 The equation of the trajectory of a projectile is $y = 0.6 x - 0.017 x ^ { 2 }$, referred to horizontal and vertically upward axes through the point of projection.

Find the angle of projection of the projectile, and show that the initial speed is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the speed and direction of motion of the projectile when it is at a height of 5.2 m above the level of the point of projection for the second time.

CAIE M2 2013 June Q1

1 A small ball is projected with speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $45 ^ { \circ }$ above the horizontal from a point $O$ on horizontal ground. At time $t \mathrm {~s}$ after projection, the horizontal and vertically upwards displacements of the ball from $O$ are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Express $x$ and $y$ in terms of $t$.
Show that the equation of the trajectory of the ball is $y = x - \frac { 1 } { 40 } x ^ { 2 }$.
State the distance from $O$ of the point at which the ball first strikes the ground.

CAIE M2 2013 June Q2

2 A particle $P$ of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 19.2 N . The other end of the string is attached to a fixed point $A$. The particle $P$ is released from rest at the point 2.7 m vertically above $A$. Calculate

the initial acceleration of $P$,
the speed of $P$ when it reaches $A$.

CAIE M2 2013 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-2_373_759_1119_694} A uniform object $A B C$ is formed from two rods $A B$ and $B C$ joined rigidly at right angles at $B$. The rod $A B$ has length 0.3 m and the rod $B C$ has length 0.2 m . The object rests with the end $A$ on a rough horizontal surface and the $\operatorname { rod } A B$ vertical. The object is held in equilibrium by a horizontal force of magnitude 4 N applied at $B$ and acting in the direction $C B$ (see diagram).

Find the distance of the centre of mass of the object from $A B$.
Calculate the weight of the object.
Find the least possible value of the coefficient of friction between the surface and the object.

CAIE M2 2013 June Q4

4 A particle of mass 0.2 kg is projected vertically downwards with initial speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A resisting force of magnitude $0.09 v \mathrm {~N}$ acts vertically upwards on the particle during its descent, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the downwards velocity of the particle at time $t \mathrm {~s}$ after being set in motion.

Show that the acceleration of the particle is $( 10 - 0.45 v ) \mathrm { m } \mathrm { s } ^ { - 2 }$.
Find $v$ when $t = 1.5$.

CAIE M2 2013 June Q5

5 A particle $P$ is projected with speed $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $40 ^ { \circ }$ above the horizontal from a point $O$. For the instant 2.5 s after projection, calculate

the speed of $P$,
the angle between $O P$ and the horizontal.

CAIE M2 2013 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_259_890_584_630} One end of a light inextensible string of length 0.2 m is attached to a fixed point $A$ which is above a smooth horizontal table. A particle $P$ of mass 0.3 kg is attached to the other end of the string. $P$ moves on the table in a horizontal circle, with the string taut and making an angle of $60 ^ { \circ }$ with the downward vertical (see diagram).

Calculate the tension in the string if the speed of $P$ is $1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
For the motion as described, show that the angular speed of $P$ cannot exceed $10 \mathrm { rad } \mathrm { s } ^ { - 1 }$, and hence find the greatest possible value for the kinetic energy of $P$.

CAIE M2 2013 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_519_860_1430_641}
$O A B C$ is the cross-section through the centre of mass of a uniform prism of weight 20 N . The crosssection is in the shape of a sector of a circle with centre $O$, radius $O A = r \mathrm {~m}$ and angle $A O C = \frac { 2 } { 3 } \pi$ radians. The prism lies on a plane inclined at an angle $\theta$ radians to the horizontal, where $\theta < \frac { 1 } { 3 } \pi$. OC lies along a line of greatest slope with $O$ higher than $C$. The prism is freely hinged to the plane at $O$. A force of magnitude 15 N acts at $A$, in a direction towards to the plane and at right angles to it (see diagram). Given that the prism remains in equilibrium, find the set of possible values of $\theta$.

CAIE M2 2013 June Q1

1 A small sphere of mass 0.4 kg moves with constant speed $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a horizontal circle inside a smooth fixed hollow cylinder of diameter 0.6 m . The axis of the cylinder is vertical, and the sphere is in contact with both the horizontal base and the vertical curved surface of the cylinder.

Calculate the magnitude of the force exerted on the sphere by the vertical curved surface of the cylinder.
Hence show that the magnitude of the total force exerted on the sphere by the cylinder is 5 N .

CAIE M2 2013 June Q2

2 A uniform semicircular lamina of radius 0.25 m has diameter $A B$. It is freely suspended at $A$ from a fixed point and hangs in equilibrium.

Find the distance of the centre of mass of the lamina from the diameter $A B$.
Calculate the angle which the diameter $A B$ makes with the vertical. The lamina is now held in equilibrium with the diameter $A B$ vertical by means of a force applied at $B$. This force has magnitude 6 N and acts at $45 ^ { \circ }$ to the upward vertical in the plane of the lamina.
Calculate the weight of the lamina.

CAIE M2 2013 June Q3

3 A particle $P$ of mass 0.2 kg is attached to one end of a light elastic string of natural length 1.6 m and modulus of elasticity 18 N . The other end of the string is attached to a fixed point $O$ which is 1.6 m above a smooth horizontal surface. $P$ is placed on the surface vertically below $O$ and then projected horizontally. $P$ moves with initial speed $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a straight line on the surface. Show that, when $O P = 1.8 \mathrm {~m}$,

$P$ is at instantaneous rest,
$P$ is on the point of losing contact with the surface.

CAIE M2 2013 June Q4

4 A ball $B$ is projected from a point $O$ on horizontal ground at an angle of $40 ^ { \circ }$ above the horizontal. $B$ hits the ground 1.8 s after the instant of projection. Calculate

the speed of projection of $B$,
the greatest height of $B$,
the distance from $O$ of the point at which $B$ hits the ground.

CAIE M2 2013 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_341_529_260_808} A block $B$ of mass 3 kg is attached to one end of a light elastic string of modulus of elasticity 70 N and natural length 1.4 m . The other end of the string is attached to a particle $P$ of mass 0.3 kg . $B$ is at rest 0.9 m from the edge of a horizontal table and the string passes over a small smooth pulley at the edge of the table. $P$ is released from rest at a point next to the pulley and falls vertically. At the first instant when $P$ is 0.8 m below the pulley and descending, $B$ is in limiting equilibrium with the part of the string attached to $B$ horizontal (see diagram).

Calculate the speed of $P$ when $B$ is first in limiting equilibrium.
Find the coefficient of friction between $B$ and the table.

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