CAIE A-Level Mathematics 4.4 Newton's Laws of Motion Question Bank

Practise applying Newton's second law to particles, vehicles and connected systems with weight, reaction, tension, thrust, resistance and friction on horizontal or inclined paths.

Exam points

draw and resolve the forces on each body before writing ΣF = ma along its motion
use a shared acceleration and appropriate tension relations for connected particles
solve a whole-system equation for acceleration, then an individual equation for tension or thrust

Covered syllabus points

4.4.1—Forces 4.4.2—Mass and weight 4.4.3—Constant acceleration 4.4.4—Connected-particle models

[Maximum number: 4]

A particle P of mass 0.2 kg lies at rest on a rough horizontal plane. A horizontal force of 1.2 N is applied to P.

Given instead that the coefficient of friction between P and the plane is 0.3 , find the distance travelled by P in the third second of its motion.

$1.2-0.3 \times 0.2 g=0.2 a$

*M1

Resolve horizontally using
Newton's Second Law; 3 relevant
terms; allow sign errors; R=0.2 g
only.

a=3

A1

0.6=0.2 a only seen, allow with
BOD, but if 0.6 as friction being used as resultant force, this is
M0A0.

$s_{3}=0+\frac{1}{2} \times 3 \times 3^{2}[=13.5] s_{2}=0+\frac{1}{2} \times 3 \times 2^{2}[=6]$

DM1

For use of $s=u t+\frac{1}{2} a t^{2}$ (or a
complete method) to find a distance
at least once with u=0 and their
positive a and t=2 or t=3.

Distance $=13.5-6=7.5 \mathrm{~m}$

A1

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[Maximum number: 9]

Figure for Question 4.4 Newton's laws of motion question 2 — CAIE A-Level Mathematics A2

Two particles A and B, of masses $3 m \mathrm{~kg}$ and $2 m \mathrm{~kg}$ respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle $\theta$ to the horizontal. A lies on the plane and B hangs vertically, 0.8 m above the floor, which is horizontal. The string between A and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially A and B are at rest.

(a)

Show that the coefficient of friction between A and the plane is $\frac{1}{10} \sqrt{3}$ .

[ 5 ]

$R=3 m g \cos 30$
B1
Use of $F=\mu R$
M1
$2 m g-T=0.1 \times 2 m$ OR $T-3 m g \sin 30-\mu \times 3 m g \cos 30=0.1 \times 3 m$
M1
$2 m g-0.2 m-3 m g \sin 30-\mu \times 3 m g \cos 30=0.1 \times 3 m$
M1
$\mu=\frac{\sqrt{3}}{10}$
A1

(b)

When B reaches the floor it comes to rest.

Find the length of time after B reaches the floor for which A is moving up the plane. [You may assume that A does not reach the pulley.]

[ 4 ]

$v^{2}=0+2 \times 0.1 \times 0.8 \quad(v=0.4)$
M1
$-3 m g \sin 30-\mu \times 3 m g \cos 30=3 m a(a=-6.5)$
M1
0=-0.4-6.5 t
M1
$t=0.4 / 6.5=0.0615 \mathrm{~s}$
A1

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Covered syllabus points

4.4 Newton's laws of motion question 1

4.4 Newton's laws of motion question 2

Question (a)

Question (b)