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

Name of student | Date | ||||

Adm. number | Year/grade | 11 | 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 | 6 | 6 | 6 | 7 | 8 | 9 | 9 | 8 | 12 | 12 | 12 | 99 |

Score |

Question 1

The diagram shows three coplanar forces acting at the point $O$. The magnitudes of the forces are $6 \mathrm{~N}$, $8 \mathrm{~N}$ and $10 \mathrm{~N}$. The angle between the $6 \mathrm{~N}$ force and the $8 \mathrm{~N}$ force is $90^{\circ}$. The forces are in equilibrium. Find the other angles between the forces. $[4]$

Question 2

A car moves in a straight line with initial speed $u \mathrm{~m} \mathrm{~s}^{-1}$ and constant acceleration $a \mathrm{~m} \mathrm{~s}^{-2}$. The car takes $5 \mathrm{~s}$ to travel the first $80 \mathrm{~m}$ and it takes $8 \mathrm{~s}$ to travel the first $160 \mathrm{~m}$. Find $a$ and $u$. $[6]$

Question 3

Coplanar forces of magnitudes $12 \mathrm{~N}, 24 \mathrm{~N}$ and $30 \mathrm{~N}$ act at a point in the directions shown in the diagram.

$\text{(i)}$ Find the components of the resultant of the three forces in the $x$-direction and in the $y$-direction. $[4]$

Component in $x$-direction.

Component in $y$-direction.

$\text{(ii)}$ Hence find the direction of the resultant. $[2]$

Question 4

A train of mass $400000 \mathrm{~kg}$ is moving on a straight horizontal track. The power of the engine is constant and equal to $1500 \mathrm{~kW}$ and the resistance to the train's motion is $30000 \mathrm{~N}$. Find

$\text{(i)}$ the acceleration of the train when its speed is $37.5 \mathrm{~ms}^{-1}$, $[4]$

$\text{(ii)}$ the steady speed at which the train can move. $[2]$

Question 5

A constant resistance to motion of magnitude $350 \mathrm{~N}$ acts on a car of mass $1250 \mathrm{~kg}$. The engine of the car exerts a constant driving force of $1200 \mathrm{~N}$. The car travels along a road inclined at an angle of $\theta$ to the horizontal, where $\sin \theta=0.05$. Find the speed of the car when it has moved $100 \mathrm{~m}$ from rest in each of the following cases.

$\circ$ The car is moving up the hill.

$\circ$ The car is moving down the hill.

Question 6

A light inextensible string has a particle $A$ of mass $0.26 \mathrm{~kg}$ attached to one end and a particle $B$ of mass $0.54 \mathrm{~kg}$ attached to the other end. The particle $A$ is held at rest on a rough plane inclined at angle $\alpha$ to the horizontal, where $\sin \alpha=\frac{5}{13}$. The string is taut and parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley at the top of the plane. Particle $B$ hangs at rest vertically below the pulley (see diagram). The coefficient of friction between $A$ and the plane is $0.2$. Particle $A$ is released and the particles start to move.

$\text{(i)}$ Find the magnitude of the acceleration of the particles and the tension in the string. $[6]$

Particle $A$ reaches the pulley $0.4 \mathrm{~s}$ after starting to move.$\text{(ii)}$ Find the distance moved by each of the particles. $[2]$

Question 7

A particle travels in a straight line from a point $P$ to a point $Q$. Its velocity $t$ seconds after leaving $P$ is $v \mathrm{~m} \mathrm{~s}^{-1}$, where $v=4 t-\frac{1}{16} t^{3}$. The distance $P Q$ is $64 \mathrm{~m}$.

$\text{(i)}$ Find the time taken for the particle to travel from $P$ to $Q$. $[5]$

$\text{(ii)}$ Find the set of values of $t$ for which the acceleration of the particle is positive. $[4]$

Question 8

A particle starts from rest at a point $O$ and moves in a horizontal straight line. The velocity of the particle is $v \mathrm{~m} \mathrm{~s}^{-1}$ at time $t \mathrm{~s}$ after leaving $O$. For $0 \leqslant t<60$, the velocity is given by

$$ v=0.05 t-0.0005 t^{2} . $$The particle hits a wall at the instant when $t=60$, and reverses the direction of its motion. The particle subsequently comes to rest at the point $A$ when $t=100$, and for $60

$\text{(i)}$ Find the velocity of the particle immediately before it hits the wall, and its velocity immediately after its hits the wall. $[2]$

$\text{(ii)}$ Find the total distance travelled by the particle. $[4]$

$\text{(iii)}$ Find the maximum speed of the particle and sketch the particle's velocity-time graph for $0 \leqslant t \leqslant 100$, showing the value of $t$ for which the speed is greatest. $[4]$

Question 9

A particle $P$ moves in a straight line passing through a point $O$. At time $t \mathrm{~s}$, the acceleration, $a \mathrm{~m} \mathrm{~s}^{-2}$, of $P$ is given by $a=6-0.24 t$. The particle comes to instantaneous rest at time $t=20$.

$\text{(i)}$ Find the value of $t$ at which the particle is again at instantaneous rest. $[5]$

$\text{(ii)}$ Find the distance the particle travels between the times of instantaneous rest. $[3]$

Question 10

A small ring $R$ is attached to one end of a light inextensible string of length $70 \mathrm{~cm}$. A fixed rough vertical wire passes through the ring. The other end of the string is attached to a point $A$ on the wire, vertically above $R$. A horizontal force of magnitude $5.6 \mathrm{~N}$ is applied to the point $J$ of the string $30 \mathrm{~cm}$ from $A$ and $40 \mathrm{~cm}$ from $R$. The system is in equilibrium with each of the parts $A J$ and $J R$ of the string taut and angle $A J R$ equal to $90^{\circ}$ (see diagram).

$\text{(i)}$ Find the tension in the part $A J$ of the string, and find the tension in the part $J R$ of the string. $[5]$

The ring $R$ has mass $0.2 \mathrm{~kg}$ and is in limiting equilibrium, on the point of moving up the wire.

$\text{(ii)}$ Show that the coefficient of friction between $R$ and the wire is $0.341$, correct to 3 significant figures. $[4]$

A particle of mass $m \mathrm{~kg}$ is attached to $R$ and $R$ is now in limiting equilibrium, on the point of moving down the wire.

$\text{(iii)}$ Given that the coefficient of friction is unchanged, find the value of $m$. $[3]$

Question 11

A particle $A$ of mass $1.6 \mathrm{~kg}$ rests on a horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley $P$ fixed at the edge of the table. The other end of the string is attached to a particle $B$ of mass $2.4 \mathrm{~kg}$ which hangs freely below the pulley. The system is released from rest with the string taut and with $B$ at a height of $0.5 \mathrm{~m}$ above the ground, as shown in the diagram. In the subsequent motion $A$ does not reach $P$ before $B$ reaches the ground.

$\text{(i)}$ Given that the table is smooth, find the time taken by $B$ to reach the ground. $[5]$

$\text{(ii)}$ Given instead that the table is rough and that the coefficient of friction between $A$ and the table is $\frac{3}{8}$, find the total distance travelled by $A$. You may assume that $A$ does not reach the pulley. $[7]$

Question 12

The diagram shows a triangular block with sloping faces inclined to the horizontal at $45^{\circ}$ and $30^{\circ}$. Particle $A$ of mass $0.8 \mathrm{~kg}$ lies on the face inclined at $45^{\circ}$ and particle $B$ of mass $1.2 \mathrm{~kg}$ lies on the face inclined at $30^{\circ}$. The particles are connected by a light inextensible string which passes over a small smooth pulley $P$ fixed at the top of the faces. The parts $A P$ and $B P$ of the string are parallel to lines of greatest slope of the respective faces. The particles are released from rest with both parts of the string taut. In the subsequent motion neither particle reaches the pulley and neither particle reaches the bottom of a face.

$\text{(i)}$ Given that both faces are smooth, find the speed of $A$ after each particle has travelled a distance of $0.4 \mathrm{~m}$. $[6]$

$\text{(ii)}$ It is given instead that both faces are rough. The coefficient of friction between each particle and a face of the block is $\mu$. Find the value of $\mu$ for which the system is in limiting equilibrium. $[6]$