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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 | Total |
---|---|---|---|---|---|

Marks | 9 | 11 | 11 | 9 | 40 |

Score |

Question 1 Code: 9709/53/M/J/11/6, Topic: -

$O$ and $A$ are fixed points on a horizontal surface, with $O A=0.5 \mathrm{~m}$. A particle $P$ of mass $0.2 \mathrm{~kg}$ is projected horizontally with speed $3 \mathrm{~m} \mathrm{~s}^{-1}$ from $A$ in the direction $O A$ and moves in a straight line (see diagram). At time $t \mathrm{~s}$ after projection, the velocity of $P$ is $v \mathrm{~m} \mathrm{~s}^{-1}$ and its displacement from $O$ is $x \mathrm{~m}$. The coefficient of friction between the surface and $P$ is $0.5$, and a force of magnitude $\frac{0.4}{x^{2}} \mathrm{~N}$ acts on $P$ in the direction $P O$.

$\text{(i)}$ Show that, while the particle is in motion, $v \displaystyle\frac{\mathrm{d} v}{\mathrm{~d} x}=-\left(5+\frac{2}{x^{2}}\right)$. $[2]$

$\text{(ii)}$ Calculate the distance travelled by $P$ before it comes to rest, and show that $P$ does not subsequently move. $[7]$

Question 2 Code: 9709/51/O/N/11/6, Topic: -

A smooth bead $B$ of mass $0.3 \mathrm{~kg}$ is threaded on a light inextensible string of length $0.9 \mathrm{~m}$. One end of the string is attached to a fixed point $A$, and the other end of the string is attached to a fixed point $C$ which is vertically below $A$. The tension in the string is $T \mathrm{~N}$, and the bead rotates with angular speed $\omega \mathrm{rad} \mathrm{s}^{-1}$ in a horizontal circle about the vertical axis through $A$ and $C$.

$\text{(i)}$ Given that $B$ moves in a circle with centre $C$ and radius $0.2 \mathrm{~m}$, calculate $\omega$, and hence find the kinetic energy of $B$. $[5]$

$\text{(ii)}$ Given instead that angle $A B C=90^{\circ}$, and that $A B$ makes an angle $\tan ^{-1}\left(\frac{1}{2}\right)$ with the vertical, calculate $T$ and $\omega$. $[6]$

Question 3 Code: 9709/52/O/N/11/6, Topic: -

Question 4 Code: 9709/53/O/N/11/6, Topic: -

A uniform solid consists of a hemisphere with centre $O$ and radius $0.6 \mathrm{~m}$ joined to a cylinder of radius $0.6 \mathrm{~m}$ and height $0.6 \mathrm{~m}$. The plane face of the hemisphere coincides with one of the plane faces of the cylinder.

$\text{(i)}$ Calculate the distance of the centre of mass of the solid from $O$. $[4]$

$\big[$The volume of a hemisphere of radius $r$ is $\frac{2}{3} \pi r^{3}.\big]$

$\text{(ii)}$

A cylindrical hole, of length $0.48 \mathrm{~m}$, starting at the plane face of the solid, is made along the axis of symmetry (see diagram). The resulting solid has its centre of mass at $O$. Show that the area of the cross-section of the hole is $\frac{3}{16} \pi \mathrm{m}^{2}$. $[4]$

$\text{(iii)}$ It is possible to increase the length of the cylindrical hole so that the solid still has its centre of mass at $O$. State the increase in the length of the hole. $[1]$