DEPARTMENT OF MATHEMATICS

Introduction to A level Mathematics

Mechanics course workbook

2015-2016

Name: _____________________________________

Thank you for choosing to continue to study Mathematics in the sixth form at Caedmon College Whitby. The Mathematics Department is committed to ensuring that you make good progress throughout your A level course. We hope that you will use this workbook to give you an organised set of notes for revision and that it will help you enjoy and benefit from the course more. ORGANISATION Keep this workbook, worksheets, homework and assessments in a file + BOOK SCRUTINY You will be asked to hand this in/bring it along to a progress interview with

your teacher or Mr Gower during the year TEXTBOOKS This workbook should be your main resource so we do not use a textbook in

lessons. Resources are also available in the Learning Centre, but if you wish to use textbooks at home for homework and revision, then we recommend that you buy the course textbook: Pearson Modular Mathematics for Edexcel AS and A-level, Mechanics 1.

CALCULATOR The recommended calculator for the course is the Casio FX 85GT PLUS. If you wish to use a graphical calculator, make sure it is an approved one for exams

ATTENDANCE We expect maximum attendance otherwise your place on the course may be at risk. Should you be unable to attend a lesson, for whatever reason, you MUST catch up on the work missed. You will need to copy someone else’s notes and find out what exercises have been set. Where possible this should be done before the next lesson.

ASSESSMENT Each chapter has a test and resit test. If you are falling behind in work and tests, a meeting will be called to discuss your progress. We expect all pupils to meet deadlines and maintain a minimum score close to their target grade in all assessments.

If you are falling behind in work and tests, a meeting will be called to discuss your progress. We expect all pupils to maintain a minimum of close to their target grade in all assessments.

REVISION BOOKLET We will produce revision a revision programme once you have completed TRIAL EXAM all or most of the Mechanics course. As last year there will be an expectation

that you do one chapter a week/fortnight to prepare for trial and actual exams. There will be one opportunity to resit the trial exam

You may also find the following useful: For resources and links to video tutorials; exam papers and solutions. Use the Mathematics department website:

ccwmaths.wordpress.com

And CCW Maths on twitter: @ccwmaths

Specifications, the formula book given in exams and past papers can be all downloaded from the EdExcel website

www.edexcel.com

Mr J Gower Head of KS5 Mathematics

MECHANICS 1 FORMULA SHEET

Velocity = change of displacement ; Acceleration = change of velocity time time Kinematics: (for constant acceleration)

atuv asuv 222 2

21 atuts

2

21 atvts tvus )(

21

u= initial velocity, v = final velocity, s = distance, a = acceleration, t = time Momentum = mv Impulse, I = mv – mu (= change in momentum) Impulse = force × time = F × t = Ft

So Ft = mv – mu (where m = mass, u = initial velocity, v = final velocity, F = force, t = time) Principle of conservation of momentum:

22112211 vmvmumum

OR: total momentum before impact = total momentum after impact Friction, F ≤ μR , where: μ = coefficient of friction and R = normal contact force (remember, if the object is moving or about to move, then F = μR, in fact, you can always use this formula for Mechanics 1) Force, F = ma (i.e. force = mass × acceleration) Resolving forces: From the diagram on the right, the effect of the mass, m, of the particle parallel to the plane = cosmg ,

and the effect of the mass of the particle perpendicular to the plane = sinmg .

Vectors:

Always of the form: 𝒂𝒊 + 𝒃𝒋 = (𝒂𝒃)

where: i is usually the vector reading horizontally left to right j is usually the vector reading vertically bottom to top Moments: The moment of a force about a point = force × perpendicular distance Moment = Fd

m

ɵ cosmg

sinmg

Contents

1. Kinematics

2. Vectors

3. Statics

4. Dynamics

5. Momentum

6. Moments

7. Revision

8.

M1WB: Kinematics

page

Suvat Notes BAT derive the suvat equations

BAT apply them to solve problems

BAT change between velocity measures

M1WB: Kinematics

page

WB1 A train starts from rest at a station and moves with constant acceleration. 35

seconds later it passes a signal box with a speed of 50 ms-1

What is its acceleration in ms-2 ?

WB2 A boat accelerates from 4 ms-1 to 6.5 ms-1 over 8 seconds

How far has the boat gone in this time?

M1WB: Kinematics

page

WB3 A space shuttle travelling at 20000 ms-1 slows with a deceleration of 45 ms-2 for

2 minutes. How many km will it have travelled?

WB4 A car is following a Lorry going at 60 kmph on the motorway. The car

constantly accelerates at 3 ms-2 to overtake the lorry and keeps accelerating for

the 120 metres it takes to overtake. How fast is the car travelling once it has

overtaken the lorry?

Give your answer in kmph

M1WB: Kinematics

page

M1WB: Kinematics

page

Vertical motion Notes

BAT Set up a model of vertical motion with acceleration due to gravity

BAT Find the high point of a thrown object or distance travelled

BAT Solve extended problems

M1WB: Kinematics

page

WB5 A marble falls off a shelf which is 1.6 m above the floor.

Find:

a) the time it takes to reach the floor

b) The speed with which it will reach the floor

WB6 A marble is catapulted vertically upwards with a velocity of 24.5 ms-1.

Modelling the marble as a particle moving under gravity alone, find for how

long its height exceeds 29.4 m

M1WB: Kinematics

page

WB7 A particle P is projected upwards from a point O

with a speed of 28 ms-1. Find:

a) The greatest height above O reached by P

b) The total time before P returns to O

c) The total distance travelled by the particle on return to O

M1WB: Kinematics

page

M1WB: Kinematics

page

Kinematics graphs notes

BAT Draw accurate diagrams for distance-time and velocity time graphs

BAT Find Area under a speed-time graph, gradient of a speed-time graph

BAT Explore other graphs and links

M1WB: Kinematics

page

WB8 A cyclist travels on a straight road over a 11 s period.

For the first four seconds they travel at constant speed of 8 ms-1

For the rest of the time they accelerate at a constant rate of 1 ms-2

Draw a Velocity time graph of their journey and work out the distance they have

travelled

M1WB: Kinematics

page

WB9 A man is jogging along a straight road at a constant speed of 4ms-1. He passes a

friend with a bicycle who is standing at the side of the road and 20s later cycles

to catch him up. The cyclist accelerates at a constant rate of 3 ms-2 until he

reaches a speed of 12 ms-1 . He then maintains a constant speed.

a) On the same diagram sketch the speed-time graphs of both cyclist and

jogger

b) Find the time that elapses before the cyclist meets the jogger

M1WB: Kinematics

page

WB10

Acceleration

graph

The acceleration-time graph models the motion of a particle.

At time t = 0 the particle has a velocity of 8 ms-1 in the positive direction

a) Find the velocity of the particle when t = 2

b) At what time does the particle start travelling in the negative direction?

M1WB: Vectors

page

Vector Geometry Notes

BAT apply the rules of vector geometry to solve problems

M1WB: Vectors

page

WB1

OPQ is a triangle

R is the midpoint of OP

S is the midpoint of PQ

𝑂𝑃⃗⃗⃗⃗ ⃗ = 𝑝 and 𝑂𝑄⃗⃗⃗⃗⃗⃗ = 𝑞

a) Find 𝑂𝑆⃗⃗⃗⃗ ⃗ in terms of p and q

b) Show that RS is parallel to OQ

WB2

OPQ is a triangle. T is the point on PQ for which 𝑃𝑇: 𝑇𝑄 𝑖𝑠 2: 1

𝑂𝑃⃗⃗⃗⃗ ⃗ = 𝑎 and 𝑂𝑄⃗⃗⃗⃗⃗⃗ = 𝑏

a) Write down in terms of a and b, an expression for 𝑃𝑄⃗⃗⃗⃗ ⃗

b) Express 𝑂𝑇⃗⃗⃗⃗ ⃗ in terms of a and b, Give your answer in its simplest form

p

q

P

Q

R S

O

Diagram accurately drawn

NOT

OPQ is a triangle.

R is the midpoint of OP.

S is the midpoint of PQ.

OP = p and OQ = q

Diagram accurately drawn

NOT

O P

Q

T

a

b

M1WB: Vectors

page

WB3 ABCD is a straight line

O is a point so that

𝑂𝐴⃗⃗⃗⃗ ⃗ = 𝑎 and 𝑂𝐵⃗⃗ ⃗⃗ ⃗ = 𝑏

B is the midpoint of AC

C is the midpoint of AD

Express, in terms of a and b, the vectors a) 𝐴𝐶⃗⃗⃗⃗ ⃗ b) 𝑂𝐷⃗⃗⃗⃗⃗⃗

WB4 ABCD is a parallelogram. AB is parallel to DC

and AD is parallel to BC

𝐴𝐵⃗⃗⃗⃗ ⃗ = 𝑝 and 𝐴𝐷⃗⃗ ⃗⃗ ⃗ = 𝑞

a) Express in terms of p and q: i) 𝐴𝐶⃗⃗⃗⃗ ⃗ ii) 𝐵𝐷⃗⃗⃗⃗⃗⃗

b) Express 𝐴𝑇⃗⃗⃗⃗ ⃗ in terms of p and q

ABCD is a straight line.

A B C D

a

b

O

Diagram accurately drawn

NOT

O is a point so that aOA and bOB .

B is the midpoint of AC.

C is the midpoint of AD.

Express, in terms of a and b, the vectors

(i) AC

A B

CD

p

q

ABCD is a parallelogram.

AB is parallel to DC.

AD is parallel to BC.

AB = p

AD = q

(a) Express, in terms of p and q

M1WB: Vectors

page

WB5 OABC is a parallelogram

P is the point on AC such that 𝐴𝑃 =2

3𝐴𝐶

𝑂𝐴⃗⃗⃗⃗ ⃗ = 6𝑎 and 𝑂𝐶⃗⃗⃗⃗ ⃗ = 6𝑐

a) Find the vector 𝑂𝑃⃗⃗⃗⃗ ⃗

b) Prove that OPM is a straight line

A B

CO

P

6a

6c

Diagram accurately drawn

NOT

OABC is a parallelogram.

P is the point on AC such that AP = 3

2AC.

OA = 6a. OC = 6c.

(a) Find the vector OP .

Give your answer in terms of a and c.

M1WB: Vectors

page

WB6 OPQR is a trapezium with OR parallel to

PQ

𝑂𝑃⃗⃗⃗⃗ ⃗ = 2𝑏 , 𝑃𝑄⃗⃗⃗⃗ ⃗ = 2𝑎 and 𝑂𝑅⃗⃗⃗⃗ ⃗ = 6𝑎

M is the midpoint of PQ and N is the

midpoint of OR

a) Find the vector and 𝑀𝑁⃗⃗⃗⃗⃗⃗ ⃗ in terms of a and b

X is the midpoint of MN and Y is the midpoint of QR

b) Prove that XY is parallel to OR

OPQR is a trapezium with PQ parallel to OR.

OP = 2b PQ = 2a OR = 6a

M is the midpoint of PQ and N is the midpoint of OR.

(a) Find the vector MN in terms of a and b.

M1WB: Vectors

page

Vector equations Notes BAT Find resultant vectors using vector triangles

BAT Use position & velocity vectors to describe an objects motion

BAT Find vector equations for position (and velocity)

BAT Solve problems with vector equations

M1WB: Vectors

page

WB7 Barry is at 4i + 6j relative to the origin

Barry walks over to Hannah: his change in position is 8i – 12j

Where is Hannah? If Barry took 4 seconds what are his velocity and his speed?

WB8 Barry starting position is 4i + 6j

His velocity is 2i – 3j

Where will Barry be after:

i) 10 seconds

ii) 23 seconds

Write an expression for his position after t seconds

M1WB: Vectors

page

WB9 Barry’s position is given by 4i + 6j + t(2i - 3j)

A rock has position vector 28i – 57j

When will Barry be directly North of the Rock?

When will Barry be directly East of the Rock?

WB10 Barry starts at 4i + 6j relative to the origin and walks

with constant velocity 2i - 3j

Hannah starts at 10i – 6j and walks with constant velocity 0.5i

Will they meet each other (at the same place)? If so, at what time?

M1WB: Vectors

page

WB11 At t = 0 two skaters John and Nadine have position vectors 40j and 20i

respectively, relative to the centre of the ice rink.

John has constant velocity 5i ms-1 and Nadine has constant velocity 3i + 4j

ms-1

Show that the skaters will collide and find the time of collision

M1WB: Vectors

page

WB6 At 12:00 a helicopter A sets out from its base O and flies with speed 120 kmh-

1 in the direction of vector 3i + 4j

At 12:20 helicopter B sets out from O and flies with speed 150 kmh-1 in the

direction of the vector 24i + 7j

a) Find the velocity vectors of A and B

b) Find the position vectors of A and B at 13:00

c) Calculate the distance of A from B at 13:00

At 13:30 B makes an emergency landing. A immediately changes direction

and flies at 120 kmh-1 in a straight line to B

d) Find the position vector of B from A at 13:30

e) Determine the time when A reaches B

M1WB: Statics

page

Force Diagrams Notes BAT resolve forces into i and j components

BAT add and subtract forces, finding the resultant force in different situations

BAT Use W=mg in problems

M1WB: Statics

page

WB1 A Truck is pulled by two horizontal ropes with Tension and directions shown

Find the size of the resultant force in the form xi + yj

WB2 A Sign hanging in space with mass 4 kg is pulled by two

Cables with Tension and directions shown.

Find the resultant force in the form xi + yj

10 N

8 N

400

i

10 N

600

Plan view

15 N

300

i

j

i, j notation

12 N

500

W

M1WB: Statics

page

WB3 A Sledge of mass 80 kg is being dragged by a force of 200 Newton's at an angle

of 300 to the horizontal smooth surface such that it is accelerating horizontally

forward as shownWhat is the Reaction Force?

Find the resultant force in the form xi + yj

WB4 The same Sledge (mass 80 kg) is now dragged by the same force along a rough

surface such that resistance force is 80 Newton’s

It is still accelerating horizontally forward

What is the Reaction Force?

Find the resultant force in

the form xi + yj

300

200 N

W

R

M1WB: Statics

page

M1WB: Statics

page

Equilibrium Notes

BAT Set up a model of Equilibrium of forces

BAT Use equilibrium to resolve forces and find unknowns

BAT Solve higher grade problems with algebraic constants,

simultaneous equations …

M1WB: Statics

page

WB5 The force system shown in the diagram is in equilibrium

Calculate P and Q

WB6 The force system shown in the diagram is in equilibrium

Calculate P and θ

5N

8N

P

60 ̊

θ

Q

P

8N

45 ̊

150 ̊

M1WB: Statics

page

WB7 A block with Weight = 10 N rests on a sloping surface. The surface is rough and

at an angle of 300 to the horizontal. Find the friction and reaction force

WB8 An object with Weight suspends in the air held by three inelastic strings : T1,

T2 and T3 as shown. If the Tension in T3 is 4 Newtons, find the Tension in T1

and T2

Weight

T1

T2

T

3

450

M1WB: Statics

page

WB9 An object with Weight of 10 Newtons is held at rest on a rough slope of 300 by a

force of 8 Newtons acting parallel to the slope. Friction is acting down the slope.

Find the magnitude of the reaction force and of Friction

WB10 An object is suspended in mid-air by two cables T1 and T2. T1 is at an angle of

1300 from the horizontal and has magnitude 6 Newton’s. T2 is at an angle of 700

from the horizontal. Find the magnitude of T2 and hence the magnitude of the

weight of the object

M1WB: Statics

page

Coefficient of Friction notes

BAT Understand the directly proportional relationship between Friction

force and Reaction force

BAT Solve equilibrium problems using coefficient of friction

M1WB: Statics

page

WB11 The same 5 kg sledge is put on two different surfaces pulled by 100 N force at

angle of 200. Given the values of Friction, find the resultant force and the

coefficient of friction for each surface

M1WB: Statics

page

WB12 A Block with mass 3 kg is pushed by a force parallel to the direction of the

slope .The block is at a tipping point where it is about to go up the slope.

The slope has coefficient of friction of 0.3 and is at 280 to the horizontal.

Find the values of friction and the Push force

WB13 A Block with mass 8 kg is at the point of slipping down a rough slope. The

slope is at 280 to the horizontal. Find the value of the coefficient of friction of

the slope

M1WB: Statics

page

WB14 A Block with mass 3 kg is held at rest on a rough slope by a horizontal force

(Push). The slope has coefficient of friction of 1/3 and the slope is at 300 to the

horizontal. Find the value of the Push force

WB15 A Block with Weight 10 Newton’s is held at rest on a rough slope by a

horizontal force (Push). The slope has coefficient of friction of 0.3 and the slope

is at 300 to the horizontal. Find the value of the Push, Reaction and friction

forces

M1WB: Statics

page

Connected particles in equilibrium notes

BAT Set up a model of two connected particles with uniform tension on

an inelastic string/cable

BAT Solve problems such as:

i) car towing caravan or ii) object on slope connected to hanging object

M1WB: Statics

page

WB16

WB17

M1WB: Dynamics

page

Dynamics: straight line and vertical motion Notes BAT Understand Newtons 3 laws

BAT Resolve forces for moving objects in a straight line

BAT Solve problems using Newtons 2nd law

BAT solve problems with moving objects using the suvat model

BAT Extend model to vertical motion and object on a slope

Newton’s Laws:

1st law. A particle will remain at rest or will continue to move with

constant velocity in a straight line unless acted on by a resultant force

2nd law. The force applied to a particle is proportional to the mass m of

the particle and the acceleration produced

3rd law. Every action has an equal and opposite reaction

M1WB: Dynamics

page

WB1

Sledge

A sledge of mass 600 g is dragged by forces parallel to the ground

Find the acceleration of the sledge when forces of (7i + 13j) N, (4i +4j) N

and (-2i – 5j) act on it.

Find also the magnitude and direction of the acceleration

M1WB: Dynamics

page

WB2

Car

A car travels a distance of 32m along a straight road while uniformly

accelerating from rest to 16 ms-1. By modelling the car as a particle find its

acceleration.

Given that the mass of the car is 640 Kg, find the magnitude of the accelerating

force.

WB3

Vertical

motion

A Particle of mass 2 kg is attached to the lower end of a string hanging

vertically. The particle is lowered and moves with acceleration 0.2 ms-2

Find the tension in the string

M1WB: Dynamics

page

WB4

Lift

A lift is accelerating upward at 1.5 ms-2. A child of mass 30 Kg is standing in the

lift. Treating the child as a particle find the force between the child and the floor

of the lift.

WB5

Vertical

motion

A stone of mass 0.5 kg is released from rest on the surface of the water in a

well. It takes 2 seconds to reach the bottom of the well. Given that the water

exerts a constant resistance of 2 N, find the depth of the well.

M1WB: Dynamics

page

WB6

Object on

slope find a

A parcel of mass 5 Kg is released from rest on a rough ramp of inclination

θ = 300 and slides down the ramp. The resistance due to friction is 8 N

Treating the parcel as a particle, find the acceleration of the parcel

WB7

Car up slope

WB7 A car with a constant driving force of 1500 N meets a slope inclined at an

angle of elevation of 250. The resistance forces to the car on the slope = 400 N.

If the car has weight 1800 N, find the acceleration of the car up the slope

M1WB: Dynamics

page

WB8

Car

deccelarating

A car of mass 500 kg travelling at a constant speed of 25 ms-1 reaches a slope

inclined at 300 to the horizontal. The resistance forces on the car travelling up

the slope total 300 N.

The driver takes their foot off the acceleration pedal at the start of the slope

Find the distance travelled by the car before it comes to rest

M1WB: Dynamics

page

WB9

Find angle

slope

WB 9 A parcel of mass 3 kg is sliding down a smooth inclined plane with an

acceleration of 4 ms-2 . Find the angle of inclination of the plane

WB10

Find m

WB 9 Daisy is sledging down a slope of 300 and accelerating at 1 ms-2.

The resistance force due to friction is 10 N. Find Daisy’s mass.

M1WB: Dynamics

page

M1WB: Dynamics

page

Dynamics –friction reintroduced Notes

BAT Extend model to reintroduce the coefficient of friction

BAT Solve harder problems with moving objects and several steps or

simultaneous equations;

BAT Solve problems up to exam level

M1WB: Dynamics

page

WB11

A parcel of mass 3 kg is sliding down a rough slope of inclination 300

The coefficient of friction between the parcel and the slope is 0.35. Find the

acceleration of the particle

WB12

A particle rests in limiting equilibrium on a plane inclined at 300 to the

horizontal. Determine the acceleration with which the particle will slide down

the plane when the angle of inclination is increased to 400

M1WB: Dynamics

page

WB13

Suvat

Find

coefficient

friction

A parcel of mass 5 Kg is released from rest on a rough ramp of inclination

θ = arcsin 3/5 and slides down the ramp.

After 3 secs it has a speed of 4.9 ms-1

Treating the parcel as a particle, find the coefficient of friction between the

parcel and the ramp

M1WB: Dynamics

page

WB14

Find angle of

inclination

WB14 A parcel of mass 3 kg is sliding down a rough inclined plane with an

acceleration equal to g cos ms-2 . Find the angle of inclination of the plane if

the coefficient of friction between the parcel and plane is 0.6

M1WB: Dynamics

page

Connected Particles notes

BAT Solve dynamics problems with connected particles:

i) the car and caravan problems (straight line) or towed at an angle ii) problems with pulleys

M1WB: Dynamics

page

WB15

moving in

same

direction

Two particles P and Q, of masses 5kg and 3kg respectively, are connected by a

light, inextensible string. Particle P is pulled by a horizontal force of 40N along

a rough horizontal plane. The coefficient of friction is 0.2 and the string is taut.

WB16

Tow bar

A car of mass 1000 kg tows a caravan of mass 750 kg along a straight road. The

engine of the car exerts a forward force of 2.5 kN The resistances to the motion

of the car and caravan are each k × their mass where k is a constant. Given that

the car accelerates at 1 ms-2 find the Tension in the tow-bar.

M1WB: Dynamics

page

WB17

Tow bar at

angle

This figure shows a lorry of mass 1600 kg towing a car of mass 900 kg along a

straight horizontal road. The two vehicles are joined by a light towbar which is

at an angle of 15° to the road. The lorry and the car experience constant

resistances to motion of magnitude 600 N and 300 N respectively. The lorry’s

engine produces a constant horizontal force on the lorry of magnitude 1500 N.

Find

(a) the acceleration of the lorry and the car, (3)

b) the tension in the towbar. (4)

Taken

further

When the speed of the vehicles is 6 m s–1, the towbar breaks. Assuming that the

resistance to the motion of the car remains of constant magnitude 300 N,

c) find the distance moved by the car from the moment the towbar breaks to the

moment when the car comes to rest. (4)

d) State whether, when the towbar breaks, the normal reaction of the road on the car

is increased, decreased or remains constant. Give a reason for your answer (2)

M1WB: Dynamics

page

WB18

Tow bar on

slope

A lorry of mass 900 kg is towing a trailer of mass 500 kg up an inclined road, at

angle α, where tan α = ¾ .The two vehicles are joined by a light towbar . The

lorry and the trailer experience constant resistances to motion of magnitude 1600

N and 600 N respectively. The lorry’s engine produces a constant horizontal

force on the lorry of magnitude 12000 N. Find

a) the acceleration of the lorry and car, (3)

b) the tension in the towbar. (4)

WB19

Tow bar -

decceleration

A lorry of mass 900 kg is towing a trailer of mass 300 kg along a straight road.

The two vehicles are joined by a light towbar . Assume resistance forces are

negligible. The driver sees a red light ahead and brakes causing a braking force

of 2400 N. Find

a) the tension in the towbar, (4)

b) the distance travelled before stopping (4)

M1WB: Dynamics

page

WB20

Different

directions

Two particles of mass 2m and 3m are connected by a light, inextensible string

over a smooth pulley

Find the acceleration of the particles. Find the tension in the string

Find the force exerted by the string on the pulley

WB21

Pulley – table

+ suspended

weight

Two particles P and Q of masses 6 kg and 3 kg are connected by a light

inextensible string. Particle P rests on a rough horizontal table. The string passes

over a smooth pulley fixed at the edge of the table and Q hangs vertically. The

system starts from rest.

If the coefficient of friction μ = 1/3 , find a) The acceleration of Q

b) The Tension in the string c) The force exerted on the pulley

M1WB: Dynamics

page

WB22

Pulley –

Slope +

suspended

weight

A particle P of mass 5 kg lies on a smooth inclined plane of angle θ = arcsin 3/5 .

Particle P is connected to a particle Q of mass 4 kg by a light inextensible string

which lies along a line of greatest slope on the plane and passes over a smooth

peg. The system is held at rest with Q hanging vertically 2 m above a horizontal

plane. The system is now released from rest. Assuming P does not reach the peg,

find, to 3sf: a) The acceleration of Q b) How long t takes for Q to hit the

horizontal plane c) the total distance that P moves up the plane.

M1WB: Dynamics

page

WB23

Pulley – table

+ suspended

weight

A particle A of mass 0.8 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 1.2 kg which hangs freely below the pulley, as shown in the diagram

above. The system is released from rest with the string taut and with B at a

height of 0.6 m above the ground. In the subsequent motion A does not reach P

before B reaches the ground. In an initial model of the situation, the table is

assumed to be smooth. Using this model, find

a) the tension in the string before B reaches the ground,

b) the time taken by B to reach the ground

In a refinement of the model, it is assumed that the table is rough and that the

coefficient of friction between A and the table is 1/5

Using this refined model,

(c) find the time taken by B to reach the ground.

M1WB: Dynamics

page

WB24

Pulley –

Slope +

suspended

weight

A particle A of mass 4 kg moves on the inclined face of a smooth wedge.

This face is inclined at 30° to the horizontal. The wedge is fixed on horizontal ground.

Particle A is connected to a particle B, of mass 3 kg, by a light inextensible string. The

string passes over a small light smooth pulley which is fixed at the top of the plane. The

section of the string from A to the pulley lies in a line of greatest slope of the wedge.

The particle B hangs freely below the pulley, as shown in the diagram above. The

system is released from rest with the string taut. For the motion before A reaches the

pulley and before B hits the ground, find

a) the tension in the string,

b) the magnitude of the resultant force exerted by the string on the pulley.

c) The string in this question is described as being ‘light’.

(i) Write down what you understand by this description.

(ii) State how you have used the fact that the string is light in your answer to part (a).

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WB25 Pulley

– Slope +

suspended

weight

The diagram above shows two particles A and B, of mass m kg and 0.4 kg

respectively, connected by a light inextensible string.

Initially A is held at rest on a fixed smooth plane inclined at 30° to the

horizontal.

The string passes over a small light smooth pulley P fixed at the top of the plane.

The section of the string from A to P is parallel to a line of greatest slope of the

plane. The particle B hangs freely below P. The system is released from rest

with the string taut and B descends with acceleration g.

a) Write down an equation of motion for B

b) Find the tension in the string

c) Prove that m = 35

16

d) State where in the calculations you have used the information that P is a

light smooth pulley.

On release, B is at a height of one metre above the ground and AP = 1.4

m.

The particle B strikes the ground and does not rebound.

e) Calculate the speed of B as it reaches the ground.

f) Show that A comes to rest as it reaches P

M1WB: Moments

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Moments on rods and beams - Notes BAT find the moment on object at a pivot;

BAT understand notation and units for moments

M1WB: Moments

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WB1 A light rod AB is 4 m long and can rotate in a vertical plane about fixed point C

Where AC = 1 m. A vertical force F of 8 N acts on the rod downwards.

Find the moment of F about C when F acts a) at A b) at B c) at C

M1WB: Moments

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WB2 The diagram shows forces acting on a lamina. Find the sum of the moments

acting about point O and state the sense of the moment (cw or acw)

WB3 A uniform rod AB of weight 20 N is 4 m long and can rotate in a vertical plane

about fixed point C. Where AC = 1 m. A vertical force F of 8 N acts on the rod

downwards at A and a vertical force of 12 N acts upwards at B.

Find the sum of moments about C

A = 12 N

2 m

. 2 m O

1.5 m B = 8 N

LAMINA

M1WB: Moments

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M1WB: Moments

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Moments on rods and beams - Notes BAT find the moment on a rod or beam at a pivot;

BAT use equilibrium to solve problems with moments and forces

M1WB: Moments

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WB4 A uniform rod AB of length 4 m and mass 5 kg is pivoted at C where AC = 1.5 m

Calculate the mass of the particle which must be attached at A to maintain equilibrium

with the rod horizontal.

WB5 A uniform beam AB of length 5 m and mass 30 kg rests horizontally on supports

at C and D. Where AC = BD = 1 m. A man of mass 75 kg stands on the beam at

E where AE = 2 m.

Calculate the magnitude of the reaction at each of the supports C and D.

M1WB: Moments

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WB6 A uniform plank AB has mass 40 kg and length 4 m.

It is supported in a horizontal position by two smooth pivots, one at the end A,

the other at the point C of the plank where AC = 3 m, as shown in the diagram

above.

A man of mass 80 kg stands on the plank which remains in equilibrium.

The magnitudes of the reactions at the two pivots are each equal to R newtons.

By modelling the plank as a rod and the man as a particle, find

a) the value of R,

b) the distance of the man from A

M1WB: Moments

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M1WB: Moments

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Non uniform rods and Tilting Notes BAT include idea of a tilting point in the model

BAT extend model to include non-uniform rods

M1WB: Moments

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WB7 A plank AE, of length 6 m and mass 10 kg, rests in a horizontal position on supports

at B and D, where AB = 1 m and DE = 2 m.

A child of mass 20 kg stands at C, the mid-point of BD, as shown in the diagram.

The child is modeled as a particle and the plank as a uniform rod.

The child and the plank are in equilibrium. Calculate:

a) the magnitude of the force exerted by the support on the plank at B,

b) the magnitude of the force exerted by the support on the plank at D.

The child now stands at a point F on the plank.

The plank is in equilibrium and on the point of tilting about D.

c) Calculate the distance DF.

M1WB: Moments

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WB8

M1WB: Moments

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WB9 A plank AB has mass 40 kg and length 3 m.

A load of mass 20 kg is attached to the plank at B.

The loaded plank is held in equilibrium, with AB horizontal, by two vertical

ropes attached at A and C, as shown in the diagram.

The plank is modelled as a uniform rod and the load as a particle.

Given that the tension in the rope at C is three times the tension in the rope at A,

calculate

a) the tension in the rope at C,

b) the distance CB

M1WB: Moments

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WB10 A steel girder AB has weight 210 N. It is held in equilibrium in a horizontal

position by two vertical cables. One cable is attached to the end A. The other

cable is attached to the point C on the girder, where AC = 90 cm, as shown in

the figure above. The girder is modelled as a uniform rod, and the cables as light

inextensible strings.

Given that the tension in the cable at C is twice the tension in the cable at A, find

a) the tension in the cable at A,

b) show that AB = 120 cm

A small load of weight W newtons is attached to the girder at B. The load is

modelled as a particle. The girder remains in equilibrium in a horizontal

position.

The tension in the cable at C is now three times the tension in the cable at A.

(c) Find the value of W.

M1WB: Moments

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WB11 A non-uniform rod AB has length 5 m and weight 200 N.

The rod rests horizontally in equilibrium on two smooth supports C & D.

The centre of mass of AB is x metres from A.

A particle of weight W newtons is placed on the rod at A.

The rod remains in equilibrium and the magnitude of the reaction of C on the rod is

160N

(a) Show that 50x – W = 100.

The particle is now removed from A and placed on the rod at B. The rod remains in

equilibrium and the reaction of C on the rod now has magnitude 50N.

(b) Obtain another equation connecting W and x.

c) Calculate the value of x and the value of W.

M1WB: Moments

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WB12 A uniform plank AB has mass 40 kg and length 3 m.

A load of mass 20 kg is attached to the plank at B.

The loaded plank is held in equilibrium, with AB horizontal, by two vertical ropes

attached at A and C, as shown in the diagram.

The plank is modelled as a uniform rod and the load as a particle.

Given that the tension in the rope at C is three times the tension in the rope at A,

calculate

a) the tension in the rope at C,

b) the distance CB

M1WB: Moments

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WB13 A plank AB has length 4 m. It lies on a horizontal platform, with the end A lying on the

platform and the end B projecting over the edge, as shown below.

The edge of the platform is at the point C.

Jack and Jill are experimenting with the plank.

Jack has mass 40 kg and Jill has mass 25 kg.

They discover that, if Jack stands at B and Jill stands at A and BC = 1.6 m, the plank is

in equilibrium and on the point of tilting about C.

By modelling the plank as a uniform rod, and Jack and Jill as particles,

(a) Find the mass of the plank.

They now alter the position of the plank in relation to the platform so that, when Jill

stands at B and Jack stands at A, the plank is again in equilibrium and on the point of

tilting about C.

(b) Find the distance BC in this position.

c) State how you have used the modelling assumptions that

(i) the plank is uniform,

(ii) the plank is a rod,

(iii) Jack and Jill are particles.

M1WB: Moments

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WB14 A steel girder AB has weight 210 N. It is held in equilibrium in a horizontal position by two

vertical cables. One cable is attached to the end A. The other cable is attached to the point C

on the girder, where AC = 90 cm, as shown in the figure above. The girder is modelled as a

uniform rod, and the cables as light inextensible strings.

Given that the tension in the cable at C is twice the tension in the cable at A, find

a) the tension in the cable at A,

b) show that AB = 120 cm

A small load of weight W newton’s is attached to the girder at B. The load is modelled as a

particle. The girder remains in equilibrium in a horizontal position.

The tension in the cable at C is now three times the tension in the cable at A.

c) Find the value of W.

M1WB: Moments

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M1WB: Momentum

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Notes BAT Find momentum of objects and change in their momentum

BAT Understand impulse and apply in solving problems

M1WB: Momentum

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WB1 A particle of mass 6 kg is at rest on a smooth surface.

A force of 12 N acts on the particle for 7 seconds

Find the final speed of the particle

WB2 A ball of mass 0.6 kg rebounds against a wall. The impulse of the ball on the

wall is 17 Ns and its speed is 15 ms-1 immediately before hitting the wall

Find the speed of the ball immediately after the rebound.

M1WB: Momentum

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Notes

BAT Find momentum of objects and change in their momentum

BAT Understand impulse and apply in solving problems

BAT Understand conservation of momentum and apply to solve different

types of problems involving collisions

M1WB: Momentum

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WB3 Particle A with mass 500 kg moving with velocity 10 ms-1 hits particle B with

mass 800 kg which is at rest. Particle A rebounds with velocity 4 ms-1.

i) What is the velocity of particle B after the collision?

ii) What is the change in momentum of each particle?

WB4 A snooker ball P moving with speed 4 ms-1 hits a stationary ball Q of equal

mass. After the impact both balls move in the same direction along the same

straight line, but the speed of Q is twice that of P. By modelling the balls as

particles moving on a smooth horizontal surface find the speeds of the balls

M1WB: Momentum

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WB5 Particle A with mass 500 kg moving with velocity 10 ms-1 hits particle B with

mass 800 kg which is moving in the same direction but with velocity 4 ms-1 The

two Particles join together after the collision

i) What is the velocity of the joint particles after the collision?

ii) What is the change in momentum of A?

WB6 Two particles P and Q of masses 3 kg and 6 kg respectively are connected by a

light inextensible string. Initially they are at rest on a smooth table with the

string slack. Q is projected directly away from P with a speed of 3 ms-1

Find their common speed when the string becomes taught

M1WB: Momentum

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WB7 A missile of mass 400 kg travelling at 200 ms-1 separates into parts A and B.

As shown in the diagram A has a mass of 50 kg and B a mass of 350 kg.

After separation the speed of B is 250 ms-1 in the original direction of motion.

a) Calculate the impulse acting on B in the separation

b) Calculate the velocity of A after the separation