$\require{\cancel}$ $\require{\stix[upint]}$

Name of student | Date | ||||

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

Subject | Mechanics 1 (M1) | Variant(s) | P41, P42, P43 | ||

Start time | Duration | Stop time |

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

Marks | 4 | 5 | 6 | 7 | 6 | 10 | 7 | 11 | 9 | 9 | 14 | 10 | 98 |

Score |

Question 1 Code: 9709/43/M/J/16/1, Topic: -

A particle of mass $8 \mathrm{~kg}$ is pulled at a constant speed a distance of $20 \mathrm{~m}$ up a rough plane inclined at an angle of $30^{\circ}$ to the horizontal by a force acting along a line of greatest slope.

$\text{(i)}$ Find the change in gravitational potential energy of the particle. $[2]$

$\text{(ii)}$ The total work done against gravity and friction is $1146 \mathrm{~J}$. Find the frictional force acting on the particle. $[2]$

Question 2 Code: 9709/43/M/J/14/2, Topic: -

A car of mass $1250 \mathrm{~kg}$ travels up a straight hill inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha=0.02$. The power provided by the car's engine is $23 \mathrm{~kW}$. The resistance to motion is constant and equal to $600 \mathrm{~N}$. Find the speed of the car at an instant when its acceleration is $0.5 \mathrm{~m} \mathrm{~s}^{-2}$. $[5]$

Question 3 Code: 9709/43/M/J/21/3, Topic: -

Four coplanar forces act at a point. The magnitudes of the forces are $20 \mathrm{~N}, 30 \mathrm{~N}, 40 \mathrm{~N}$ and $F \mathrm{~N}$. The directions of the forces are as shown in the diagram, where $\sin \alpha^{\circ}=0.28$ and $\sin \beta^{\circ}=0.6$.

Given that the forces are in equilibrium, find $F$ and $\theta$. $[6]$

Question 4 Code: 9709/43/M/J/11/4, Topic: -

The diagram shows the velocity-time graphs for the motion of two particles $P$ and $Q$, which travel in the same direction along a straight line. $P$ and $Q$ both start at the same point $X$ on the line, but $Q$ starts to move $T \mathrm{~s}$ later than $P$. Each particle moves with speed $2.5 \mathrm{~m} \mathrm{~s}^{-1}$ for the first $20 \mathrm{~s}$ of its motion. The speed of each particle changes instantaneously to $4 \mathrm{~m} \mathrm{~s}^{-1}$ after it has been moving for $20 \mathrm{~s}$ and the particle continues at this speed.

$\text{(i)}$ Make a rough copy of the diagram and shade the region whose area represents the displacement of $P$ from $X$ at the instant when $Q$ starts. $[1]$

It is given that $P$ has travelled $70 \mathrm{~m}$ at the instant when $Q$ starts.

$\text{(ii)}$ Find the value of $T$. $[2]$

$\text{(iii)}$ Find the distance between $P$ and $Q$ when $Q$ 's speed reaches $4 \mathrm{~m} \mathrm{~s}^{-1}$. $[2]$

$\text{(iv)}$ Sketch a single diagram showing the displacement-time graphs for both $P$ and $Q$, with values shown on the $t$-axis at which the speed of either particle changes. $[2]$

Question 5 Code: 9709/43/M/J/15/4, Topic: -

A lorry of mass $12000 \mathrm{~kg}$ moves up a straight hill of length $500 \mathrm{~m}$, starting at the bottom with a speed of $24 \mathrm{~m} \mathrm{~s}^{-1}$ and reaching the top with a speed of $16 \mathrm{~m} \mathrm{~s}^{-1}$. The top of the hill is $25 \mathrm{~m}$ above the level of the bottom of the hill. The resistance to motion of the lorry is $7500 \mathrm{~N}$. Find the driving force of the lorry. $[6]$

Question 6 Code: 9709/43/M/J/19/4, Topic: -

Two particles $A$ and $B$, of masses $1.3 \mathrm{~kg}$ and $0.7 \mathrm{~kg}$ respectively, are connected by a light inextensible string which passes over a smooth fixed pulley. Particle $A$ is $1.75 \mathrm{~m}$ above the floor and particle $B$ is $1 \mathrm{~m}$ above the floor (see diagram). The system is released from rest with the string taut, and the particles move vertically. When the particles are at the same height the string breaks.

$\text{(i)}$ Show that, before the string breaks, the magnitude of the acceleration of each particle is $3 \mathrm{~m} \mathrm{~s}^{-2}$ and find the tension in the string. $[4]$

$\text{(ii)}$ Find the difference in the times that it takes the particles to hit the ground. $[6]$

Question 7 Code: 9709/42/M/J/13/5, Topic: -

A car of mass $1000 \mathrm{~kg}$ is travelling on a straight horizontal road. The power of its engine is constant and equal to $P \mathrm{~kW}$. The resistance to motion of the car is $600 \mathrm{~N}$. At an instant when the car's speed is $25 \mathrm{~m} \mathrm{~s}^{-1}$, its acceleration is $0.2 \mathrm{~m} \mathrm{~s}^{-2}$. Find

$\text{(i)}$ the value of $P$, $[4]$

$\text{(ii)}$ the steady speed at which the car can travel. $[3]$

Question 8 Code: 9709/42/M/J/21/5, Topic: -

A car of mass $1250 \mathrm{~kg}$ is pulling a caravan of mass $800 \mathrm{~kg}$ along a straight road. The resistances to the motion of the car and caravan are $440 \mathrm{~N}$ and $280 \mathrm{~N}$ respectively. The car and caravan are connected by a light rigid tow-bar.

$\text{(a)}$ The car and caravan move along a horizontal part of the road at a constant speed of $30 \mathrm{~ms}^{-1}$.

$\text{(i)}$ Calculate, in $\mathrm{kW}$, the power developed by the engine of the car. $[2]$

$\text{(ii)}$ Given that this power is suddenly decreased by $8 \mathrm{~kW}$, find the instantaneous deceleration of the car and caravan and the tension in the tow-bar. $[4]$

$\text{(b)}$ The car and caravan now travel along a part of the road inclined at $\sin ^{-1} 0.06$ to the horizontal. The car and caravan travel up the incline at constant speed with the engine of the car working at $28 \mathrm{~kW}.$

$\text{(i)}$ Find this constant speed. $[3]$

$\text{(ii)}$ Find the increase in the potential energy of the caravan in one minute. $[2]$

Question 9 Code: 9709/43/M/J/13/6, Topic: -

A small box of mass $40 \mathrm{~kg}$ is moved along a rough horizontal floor by three men. Two of the men apply horizontal forces of magnitudes $100 \mathrm{~N}$ and $120 \mathrm{~N}$, making angles of $30^{\circ}$ and $60^{\circ}$ respectively with the positive $x$-direction. The third man applies a horizontal force of magnitude $F \mathrm{~N}$ making an angle of $\alpha^{\circ}$ with the negative $x$-direction (see diagram). The resultant of the three horizontal forces acting on the box is in the positive $x$-direction and has magnitude $136 \mathrm{~N}$.

$\text{(i)}$ Find the values of $F$ and $\alpha$. $[6]$

$\text{(ii)}$ Given that the box is moving with constant speed, state the magnitude of the frictional force acting on the box and hence find the coefficient of friction between the box and the floor. $[3]$

Question 10 Code: 9709/41/M/J/21/6, Topic: -

Three coplanar forces of magnitudes $10 \mathrm{~N}, 25 \mathrm{~N}$ and $20 \mathrm{~N}$ act at a point $O$ in the directions shown in the diagram.

$\text{(a)}$ Given that the component of the resultant force in the $x$-direction is zero, find $\alpha$, and hence find the magnitude of the resultant force. $[4]$

$\text{(b)}$ Given instead that $\alpha=45$, find the magnitude and direction of the resultant of the three forces. $[5]$

Question 11 Code: 9709/42/M/J/18/7, Topic: -

As shown in the diagram, a particle $A$ of mass $1.6 \mathrm{~kg}$ lies on a horizontal plane and a particle $B$ of mass $2.4 \mathrm{~kg}$ lies on a plane inclined at an angle of $30^{\circ}$ to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley $P$ fixed at the top of the inclined plane. The distance $A P$ is $2.5 \mathrm{~m}$ and the distance of $B$ from the bottom of the inclined plane is $1 \mathrm{~m}$. There is a barrier at the bottom of the inclined plane preventing any further motion of $B$. The part $B P$ of the string is parallel to a line of greatest slope of the inclined plane. The particles are released from rest with both parts of the string taut.

$\text{(i)}$ Given that both planes are smooth, find the acceleration of $A$ and the tension in the string. $[5]$

$\text{(ii)}$ It is given instead that the horizontal plane is rough and that the coefficient of friction between $A$ and the horizontal plane is $0.2$. The inclined plane is smooth. Find the total distance travelled by $A$. $[9]$

Question 12 Code: 9709/42/M/J/21/7, Topic: -

A particle $P$ moving in a straight line starts from rest at a point $O$ and comes to rest $16 \mathrm{~s}$ later. At time $t \mathrm{~s}$ after leaving $O$, the acceleration $a \mathrm{~m} \mathrm{~s}^{-2}$ of $P$ is given by

$$ \begin{array}{ll} a=6+4 t & 0 \leqslant t < 2 \\ a=14 & 2 \leqslant t < 4 \\ a=16-2 t & 4 \leqslant t \leqslant 16 \end{array} $$There is no sudden change in velocity at any instant.

$\text{(a)}$ Find the values of $t$ when the velocity of $P$ is $55 \mathrm{~m} \mathrm{~s}^{-1}$. $[5]$

$\text{(b)}$ Complete the sketch of the velocity-time diagram. $[2]$

$\text{(c)}$ Find the distance travelled by $P$ when it is decelerating. $[3]$