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MATHEMATICS 9709

Cambridge International AS and A Level

Name of student Date
Adm. number Year/grade Stream
Subject Mechanics 1 (M1) Variant(s) P41, P43
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 Total
Marks 3 7 6 6 6 9 12 49
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject.
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Question 1 Code: 9709/41/M/J/21/1, Topic: -

A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass $50 \mathrm{~kg}$ up a line of greatest slope of a plane inclined at $60^{\circ}$ to the horizontal. The winch pulls the load a distance of $5 \mathrm{~m}$ up the plane at constant speed. There is a constant resistance to motion of $100 \mathrm{~N}$.

Find the work done by the winch. $[3]$

Question 2 Code: 9709/43/M/J/10/3, Topic: -

A load is pulled along a horizontal straight track, from $A$ to $B$, by a force of magnitude $P \mathrm{~N}$ which acts at an angle of $30^{\circ}$ upwards from the horizontal. The distance $A B$ is $80 \mathrm{~m}$. The speed of the load is constant and equal to $1.2 \mathrm{~m} \mathrm{~s}^{-1}$ as it moves from $A$ to the mid-point $M$ of $A B$.

$\text{(i)}$ For the motion from $A$ to $M$ the value of $P$ is $25.$ Calculate the work done by the force as the load moves from $A$ to $M$. $[2]$

The speed of the load increases from $1.2 \mathrm{~m} \mathrm{~s}^{-1}$ as it moves from $M$ towards $B$. For the motion from $M$ to $B$ the value of $P$ is 50 and the work done against resistance is the same as that for the motion from $A$ to $M$. The mass of the load is $35 \mathrm{~kg}$.

$\text{(ii)}$ Find the gain in kinetic energy of the load as it moves from $M$ to $B$ and hence find the speed with which it reaches $B$. $[5]$

Question 3 Code: 9709/43/O/N/16/3, Topic: -

 

Particles $P$ and $Q$, of masses $7 \mathrm{~kg}$ and $3 \mathrm{~kg}$ respectively, are attached to the two ends of a light inextensible string. The string passes over two small smooth pulleys attached to the two ends of a horizontal table. The two particles hang vertically below the two pulleys. The two particles are both initially at rest, $0.5 \mathrm{~m}$ below the level of the table, and $0.4 \mathrm{~m}$ above the horizontal floor (see diagram).

$\text{(i)}$ Find the acceleration of the particles and the speed of $P$ immediately before it reaches the floor. $[4]$

$\text{(ii)}$ Determine whether $Q$ comes to instantaneous rest before it reaches the pulley directly above it. $[2]$

Question 4 Code: 9709/41/O/N/15/4, Topic: -

 

Blocks $P$ and $Q$, of mass $m \mathrm{~kg}$ and $5 \mathrm{~kg}$ respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at $35^{\circ}$ to the horizontal. Block $P$ is at rest on the plane and block $Q$ hangs vertically below the pulley (see diagram). The coefficient of friction between block $P$ and the plane is $0.2.$ Find the set of values of $m$ for which the two blocks remain at rest. $[6]$

Question 5 Code: 9709/41/O/N/20/4, Topic: -

A particle $P$ moves in a straight line. It starts from rest at a point $O$ on the line and at time $t \mathrm{~s}$ after leaving $O$ it has acceleration $a \mathrm{~m} \mathrm{~s}^{-2}$, where $a=6 t-18$.

Find the distance $P$ moves before it comes to instantaneous rest. $[6]$

Question 6 Code: 9709/43/M/J/20/6, Topic: -

A particle travels in a straight line $P Q$. The velocity of the particle $t \mathrm{~s}$ after leaving $P$ is $v \mathrm{~m} \mathrm{~s}^{-1}$, where

$$ v=4.5+4 t-0.5 t^{2} $$

$\text{(a)}$ Find the velocity of the particle at the instant when its acceleration is zero. $[3]$

The particle comes to instantaneous rest at $Q$.

$\text{(b)}$ Find the distance $P Q$. $[6]$

Question 7 Code: 9709/41/O/N/20/7, Topic: -

 

Three points $A, B$ and $C$ lie on a line of greatest slope of a plane inclined at an angle of $30^{\circ}$ to the horizontal, with $A B=1 \mathrm{~m}$ and $B C=1 \mathrm{~m}$, as shown in the diagram. A particle of mass $0.2 \mathrm{~kg}$ is released from rest at $A$ and slides down the plane. The part of the plane from $A$ to $B$ is smooth. The part of the plane from $B$ to $C$ is rough, with coefficient of friction $\mu$ between the plane and the particle.

$\text{(a)}$ Given that $\mu=\frac{1}{2} \sqrt{3}$, find the speed of the particle at $C$. $[8]$

$\text{(b)}$ Given instead that the particle comes to rest at $C$, find the exact value of $\mu$. $[4]$

Worked solutions: P1, P3 & P6 (S1)

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