ZIMSEC A Level Mechanics Paper 4 November 2004

Section b

Section b, Question 12

[1 marks]Kinematics, displacement-time graphs

The diagram shows the

(t,\ x)

graph for a particle which moves along a straight line. Its first straight section runs from

(0,\ -4)

to the corner at

(2,\ 2)

, and A is the point where the graph first crosses the t-axis. Find the coordinates of the point A.

Answer this when you sit the paper.

[1 marks]Kinematics, displacement-time graphs

The diagram shows the

(t,\ x)

graph for a particle which moves along a straight line, where x metres is the displacement of the particle from a fixed origin. Write down the final displacement of the particle, in metres.

Answer this when you sit the paper.

[1 marks]Kinematics, displacement-time graphs

The

(t,\ x)

graph for a particle moving along a straight line starts at

x=-4

m when

t=0

and finishes at

x=-2

m when

t=7

s, where x metres is the displacement from a fixed origin. Find the average velocity of the particle, in ms

^{-1}

Answer this when you sit the paper.

Section b, Question 13

[3 marks]Resultant of two forces

Two forces, P newtons and Q newtons, are inclined at an angle

\theta

to each other. When

|P|=9

and

|Q|=4

the resultant has the same magnitude as when

|P|=9

and

|Q|=8

, with

\theta

unchanged. Find the value of

\cos\theta

Answer this when you sit the paper.

[2 marks]Resultant of two forces

Two forces of magnitudes 9 N and 4 N are inclined at an angle

\theta

to each other, where

\cos\theta=-\dfrac{2}{3}

. Find the magnitude R, in newtons, of their resultant.

Answer this when you sit the paper.

Section b, Question 14

[3 marks]Projectile motion

A particle is projected from the point O with speed 8 ms

^{-1}

at an angle

\theta

above the horizontal, and it passes through the point

A(8;\ -1.81)

, distances in metres, with O as origin. Taking

g=9.81

^{-2}

, there are two possible values of

\theta

. Find the smaller one, in degrees, correct to 1 decimal place.

Answer this when you sit the paper.

[2 marks]Projectile motion

A particle is projected from O with speed 8 ms

^{-1}

at an angle

\theta

above the horizontal and passes through the point

A(8;\ -1.81)

, distances in metres. The two possible values of

\theta

are

45^{\circ}

and

32.2^{\circ}

. Which value gives the minimum time taken to reach A, and why?

A $32.2^{\circ}$ , because $8\cos\theta$ is larger, so the time $t=\sec\theta$ to cover the 8 m is smaller.
B $45^{\circ}$ , because $45^{\circ}$ always gives the fastest projectile flight.
C $45^{\circ}$ , because the vertical component $8\sin\theta$ is larger, so the particle falls to the level of A sooner.
DBoth give the same time, because both reach the same point A from the same point O.

Section b, Question 15

[2 marks]Newton's laws on an inclined plane

A particle A of mass

6m

kg lies on a plane inclined at

30^{\circ}

to the horizontal. Find, in terms of m and g, the magnitude of the normal reaction of the plane on A.

Answer this when you sit the paper.

[3 marks]Friction on an inclined plane

A particle A of mass

6m

kg lies on a rough plane inclined at

30^{\circ}

to the horizontal, and the coefficient of friction between the plane and A is 0.25. Find, in terms of m and g, the magnitude of the limiting friction force on A.

Answer this when you sit the paper.

[3 marks]Equilibrium of connected particles

A particle A of mass

6m

kg rests on a rough plane inclined at

30^{\circ}

to the horizontal, with coefficient of friction 0.25. A light inextensible string runs from A over a smooth pulley at the top of the plane to a particle B of mass

2m

kg hanging freely. The system is released from rest. Does A slide down the plane?

AYes: the limiting friction is $0.25\times2mg\cos30^{\circ}=0.433mg$ , which is less than the driving force $mg$ .
BNo: the driving force $6mg\sin30^{\circ}-2mg=mg$ is less than the limiting friction $1.299mg$ .
CYes: $6mg\sin30^{\circ}=3mg$ exceeds $2mg$ , so A slides down whatever the friction.
DNo: B is lighter than A, so the string stays slack and nothing moves.

[2 marks]Force on a pulley

A light inextensible string carrying tension T passes over a smooth pulley P fixed at the top of a plane inclined at

30^{\circ}

to the horizontal. One part of the string runs down the slope to a particle on the plane, and the other hangs vertically. What is the magnitude of the force exerted by the string on P?

A $T\sqrt{3}$
B $2T$
C $T$
D $T\sqrt{2}$

Mechanics Paper 4 November 2004

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The questions

Section b

Section b, Question 12

Section b, Question 13

Section b, Question 14

Section b, Question 15

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The answers, and why they are the answers

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