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### HENRYTAIGO

#### 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 Total
Marks 9 11 11 9 40
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 4 questions Question 1 Code: 9709/53/M/J/11/6, Topic: -$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]\$

Worked solutions: P1, P3 & P6 (S1)

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