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Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Mechanics 2 (M2) | Variant(s) | P41, P43 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 3 | 5 | 5 | 6 | 6 | 5 | 6 | 8 | 9 | 8 | 9 | 12 | 82 |

Score |

Question 1 Code: 9709/51/O/N/10/1, Topic: -

A horizontal circular disc rotates with constant angular speed $9 \mathrm{rad} \mathrm{s}^{-1}$ about its centre $O$. A particle of mass $0.05 \mathrm{~kg}$ is placed on the disc at a distance $0.4 \mathrm{~m}$ from $O$. The particle moves with the disc and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by the disc. $[3]$

Question 2 Code: 9709/53/O/N/11/2, Topic: -

An object is made from two identical uniform rods $A B$ and $B C$ each of length $0.6 \mathrm{~m}$ and weight $7 \mathrm{~N}$. The rods are rigidly joined to each other at $B$ and angle $A B C=90^{\circ}$.

$\text{(i)}$ Calculate the distance of the centre of mass of the object from $B$. $[1]$

The object is freely suspended at $A$ and a force of magnitude $F \mathrm{~N}$ is applied to the rod $B C$ at $C$. The object is in equilibrium with $A B$ inclined at $45^{\circ}$ to the horizontal.

$\text{(ii)}~~$ $\text{(a)}$

![9709-53-O-N-11-2a.PNG]

Calculate $F$ given that the force acts horizontally as shown in Fig. 1. $[2]$

$\text{(b)}$

![9709-53-O-N-11-2b.PNG]

Calculate $F$ given instead that the force acts perpendicular to the rod as shown in Fig. 2. $[2]$

Question 3 Code: 9709/53/O/N/15/2, Topic: -

One end of a light inextensible string of length $0.5 \mathrm{~m}$ is attached to a fixed point $A$. A particle $P$ of mass $0.2 \mathrm{~kg}$ is attached to the other end of the string. $P$ moves with constant speed in a horizontal circle with centre $O$ which is $0.4 \mathrm{~m}$ vertically below $A$.

$\text{(i)}$ Show that the tension in the string is $2.5 \mathrm{~N}$. $[2]$

$\text{(ii)}$ Find the speed of $P$. $[3]$

Question 4 Code: 9709/53/O/N/18/2, Topic: -

Question 5 Code: 9709/53/O/N/13/3, Topic: -

A particle $P$ of mass $0.5 \mathrm{~kg}$ moves in a horizontal circle on the smooth inner surface of a hollow cone which is fixed with its axis vertical and its vertex downwards. $P$ moves with angular speed $5 \mathrm{rad} \mathrm{s}^{-1}$ in a circle of radius $0.4 \mathrm{~m}$ (see diagram). Show that the semi-vertical angle of the cone is $45^{\circ}$ and calculate the magnitude of the force exerted on $P$ by the surface of the cone. $[6]$

Question 6 Code: 9709/51/O/N/15/3, Topic: -

A particle $P$ of mass $0.3 \mathrm{~kg}$ moves in a straight line on a smooth horizontal surface. $P$ passes through a fixed point $O$ of the line with velocity $8 \mathrm{~m} \mathrm{~s}^{-1}.$ A force of magnitude $2 x \mathrm{~N}$ acts on $P$ in the direction $P O$, where $x \mathrm{~m}$ is the displacement of $P$ from $O$.

$\text{(i)}$ Show that $v \displaystyle\frac{\mathrm{d} v}{\mathrm{~d} x}=k x$ and state the value of the constant $k$. $[2]$

$\text{(ii)}$ Find the value of $x$ at the instant when $P$ comes to instantaneous rest. $[3]$

Question 7 Code: 9709/51/O/N/19/3, Topic: -

A smooth horizontal surface has two fixed points $O$ and $A$ which are $0.8 \mathrm{~m}$ apart. A particle $P$ of mass $0.25 \mathrm{~kg}$ is projected with velocity $3 \mathrm{~m} \mathrm{~s}^{-1}$ horizontally from $A$ in the direction away from $O$. The velocity of $P$ is $v \mathrm{~m} \mathrm{~s}^{-1}$ when the displacement of $P$ from $O$ is $x \mathrm{~m}$. A force of magnitude $k v^{2} x^{-2} \mathrm{~N}$ opposes the motion of $P$.

$\text{(i)}$ Show that $\displaystyle v \displaystyle\frac{\mathrm{d} v}{\mathrm{~d} x}=-4 k v^{2} x^{-2}$. $[1]$

$\text{(ii)}$ Express $v$ in terms of $k$ and $x$. $[5]$

Question 8 Code: 9709/53/O/N/14/4, Topic: -

A particle $P$ is projected with speed $50 \mathrm{~m} \mathrm{~s}^{-1}$ at an angle of $30^{\circ}$ above the horizontal from a point $O$ on a horizontal plane.

$\text{(i)}$ Calculate the speed of $P$ when it has been in motion for $4 \mathrm{~s}$, and calculate another time at which $P$ has this speed. $[5]$

$\text{(ii)}$ Find the distance $O P$ when $P$ has been in motion for $4 \mathrm{~s}$. $[2]$

Question 9 Code: 9709/53/O/N/10/5, Topic: -

A light elastic string has natural length $2 \mathrm{~m}$ and modulus of elasticity $\lambda \mathrm{N}$. The ends of the string are attached to fixed points $A$ and $B$ which are at the same horizontal level and $2.4 \mathrm{~m}$ apart. A particle $P$ of mass $0.6 \mathrm{~kg}$ is attached to the mid-point of the string and hangs in equilibrium at a point $0.5 \mathrm{~m}$ below $A B$ (see diagram).

$\text{(i)}$ Show that $\lambda=26$. $[4]$

$P$ is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point $0.9 \mathrm{~m}$ below $A B$.

$\text{(ii)}$ Calculate the speed of projection of $P$. $[5]$

Question 10 Code: 9709/53/O/N/14/5, Topic: -

Two light elastic strings each have one end attached to a fixed horizontal beam. One string has natural length $0.6 \mathrm{~m}$ and modulus of elasticity $12 \mathrm{~N}$; the other string has natural length $0.7 \mathrm{~m}$ and modulus of elasticity $21 \mathrm{~N}$. The other ends of the strings are attached to a small block $B$ of weight $W \mathrm{~N}$. The block hangs in equilibrium $d \mathrm{~m}$ below the beam, with both strings vertical (see diagram).

$\text{(i)}$ Given that the tensions in the strings are equal, find $d$ and $W$. $[4]$

The small block is now raised vertically to the point $0.7 \mathrm{~m}$ below the beam, and then released from rest.

$\text{(ii)}$ Find the greatest speed of the block in its subsequent motion. $[4]$

Question 11 Code: 9709/51/O/N/19/6, Topic: -

$A$ and $B$ are two fixed points on a vertical axis with $A 0.6 \mathrm{~m}$ above $B$. A particle $P$ of mass $0.3 \mathrm{~kg}$ is attached to $A$ by a light inextensible string of length $0.5 \mathrm{~m}$. The particle $P$ is attached to $B$ by a light elastic string with modulus of elasticity $46 \mathrm{~N}$. The particle $P$ moves with constant angular speed $8 \mathrm{rad} \mathrm{s}^{-1}$ in a horizontal circle with centre at the mid-point of $A B$.

$\text{(i)}$ Find the speed of $P$. $[2]$

$\text{(ii)}$ Calculate the tension in the string $B P$ and hence find the natural length of this string. $[7]$

Question 12 Code: 9709/51/O/N/12/7, Topic: -

A light elastic string has natural length $3 \mathrm{~m}$ and modulus of elasticity $45 \mathrm{~N}$. A particle $P$ of weight $6 \mathrm{~N}$ is attached to the mid-point of the string. The ends of the string are attached to fixed points $A$ and $B$ which lie in the same vertical line with $A$ above $B$ and $A B=4 \mathrm{~m}$. The particle $P$ is released from rest at the point $1.5 \mathrm{~m}$ vertically below $A$.

$\text{(i)}$ Calculate the distance $P$ moves after its release before first coming to instantaneous rest at a point vertically above $B$. (You may assume that at this point the part of the string joining $P$ to $B$ is slack.) $[4]$

$\text{(ii)}$ Show that the greatest speed of $P$ occurs when it is $2.1 \mathrm{~m}$ below $A$, and calculate this greatest speed. $[5]$

$\text{(iii)}$ Calculate the greatest magnitude of the acceleration of $P$. $[3]$