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Cambridge International AS and A Level

Name of student HENRYTAIGO Date
Adm. number Year/grade HenryTaigo Stream HenryTaigo
Subject Mechanics 2 (M2) Variant(s) P41, P42, P43
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 6 8 6 8 8 6 42

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Question 1 Code: 9709/51/M/J/11/3, Topic: -


A light elastic string of natural length $1.2 \mathrm{~m}$ and modulus of elasticity $24 \mathrm{~N}$ is attached to fixed points $A$ and $B$ on a smooth horizontal surface, where $A B=1.2 \mathrm{~m}$. A particle $P$ is attached to the mid-point of the string. $P$ is projected with speed $0.5 \mathrm{~m} \mathrm{~s}^{-1}$ along the surface in a direction perpendicular to $A B$ (see diagram). $P$ comes to instantaneous rest at a distance $0.25 \mathrm{~m}$ from $A B$.

$\text{(i)}$ Show that the mass of $P$ is $0.8 \mathrm{~kg}$. $[3]$

$\text{(ii)}$ Calculate the greatest deceleration of $P$. $[3]$

Question 2 Code: 9709/52/M/J/11/3, Topic: -


A particle $P$ of mass $0.5 \mathrm{~kg}$ is attached to the vertex $V$ of a fixed solid cone by a light inextensible string. $P$ lies on the smooth curved surface of the cone and moves in a horizontal circle of radius $0.1 \mathrm{~m}$ with centre on the axis of the cone. The cone has semi-vertical angle $60^{\circ}$ (see diagram).

$\text{(i)}$ Calculate the speed of $P$, given that the tension in the string and the contact force between the cone and $P$ have the same magnitude. $[4]$

$\text{(ii)}$ Calculate the greatest angular speed at which $P$ can move on the surface of the cone. $[4]$

Question 3 Code: 9709/53/M/J/11/3, Topic: -


A smooth hemispherical shell, with centre $O$, weight $12 \mathrm{~N}$ and radius $0.4 \mathrm{~m}$, rests on a horizontal plane. A particle of weight $W \mathrm{~N}$ lies at rest on the inner surface of the hemisphere vertically below $O$. A force of magnitude $F \mathrm{~N}$ acting vertically upwards is applied to the highest point of the hemisphere, which is in equilibrium with its axis of symmetry inclined at $20^{\circ}$ to the horizontal (see diagram).

$\text{(i)}$ Show, by taking moments about $O$, that $F=16.48$ correct to 4 significant figures. $[3]$

$\text{(ii)}$ Find the normal contact force exerted by the plane on the hemisphere in terms of $W$. Hence find the least possible value of $W$. $[3]$

Question 4 Code: 9709/51/O/N/11/3, Topic: -

One end of a light elastic string of natural length $0.4 \mathrm{~m}$ and modulus of elasticity $20 \mathrm{~N}$ is attached to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass $0.25 \mathrm{~kg}. P$ hangs in equilibrium below $O$.

$\text{(i)}$ Calculate the distance $O P$. $[2]$

The particle $P$ is raised, and is released from rest at $O$.

$\text{(ii)}$ Calculate the speed of $P$ when it passes through the equilibrium position. $[3]$

$\text{(iii)}$ Calculate the greatest value of the distance $O P$ in the subsequent motion. $[3]$

Question 5 Code: 9709/52/O/N/11/3, Topic: -

Question 6 Code: 9709/53/O/N/11/3, Topic: -

A particle $P$ is projected with speed $25 \mathrm{~m} \mathrm{~s}^{-1}$ 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 upward displacements of $P$ from $O$ are $x \mathrm{~m}$ and $y \mathrm{~m}$ respectively.

$\text{(i)}$ Express $x$ and $y$ in terms of $t$ and hence show that the equation of the path of $P$ is $y=x-0.016 x^{2}$. $[4]$

$\text{(ii)}$ Calculate the horizontal distance between the two positions at which $P$ is $2.4 \mathrm{~m}$ above the ground. $[2]$

Worked solutions: P1, P3 & P6 (S1)

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