Section 10

Assignment 10

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 10.

Links to the assessment requirements

The assignment uses exam-type questions and will test your understanding of the concepts you have explored in this section. These address the following in Statistics and Mechanics part of the specification:

6.1

7.3, 7.4, 7.5

8.2, 8.4, 8.5, 8.6

9.1

How your tutor will mark your work

Your tutor will:

? check that you have answered each question ? check your workings as well as your answer ? give you feedback ? make suggestions about how you may be able to improve your work.

Are you ready to do the assignment?

Before you do the assignment, work through the topics in Section 10, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the assignment.

GCE A level (Part 2) Mathematics ? Section 10 ? Assignment 10

What to do

Answer as many questions as you can.

It is always important to show your working in your responses.

Guide time

The guide time for this assignment is 1 hour 30 minutes.

1

(a) As shown in the diagram above, forces 10 N and 5 N act along OA and OB respectively, where angle AOB = 60°.

The resultant of these two forces makes an angle of ?° with OA.

Find the value of ?. (3)

(b) The resultant of the three forces shown in the diagram above has a magnitude of R N and acts along OX. Find the values of

(i) P (ii) R (5)

2 © 2018 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 10 ? Assignment 10

2 A non-uniform plank AB has length 6 m and mass 30 kg. The plank rests in equilibrium in a horizontal position on supports at the points S and T of the plank where AS = 0.5 m and TB = 2 m.

When a block of mass M kg is placed on the plank at A, the plank remains horizontal and in equilibrium and the plank is on the point of tilting about S.

When the block is moved to B, the plank remains horizontal and in equilibrium and the plank is on the point of tilting about T.

The distance of the centre of mass of the plank from A is d metres. The block is modelled as a particle and the plank is modelled as a non-uniform rod. Find:

(a) the value of d

(b) the value of M. (7)

3 A boat B is moving with constant velocity. At noon, B is at the point with position vector (3i - 4j) km with respect to a fixed origin O. At 1430 on the same day, B is at the point with position vector (8i +11j) km. (a) Find the velocity of B, giving your answer in the form pi+qj. (3) At time t hours after noon, the position vector of B is b km.

(b) Find, in terms of t, an expression for b. (3)

Another boat C is also moving with constant velocity. The position vector of C, c km, at time t hours after noon, is given by c = (-9i + 20j) +t(6i + ?j), where ? is a constant. Given that C intercepts B,

(c) find the value of ? (5)

(d) show that, before C intercepts B, the boats are moving with the same speed. (3)

4 Two forces F1 and F2 act on a particle P.

The force F1 is given by F1 = (-i + 2j) N and F2 acts in the direction of the vector (i + j).

Given that the resultant of F1 and F2 acts in the direction of the vector (i + 3j),

(a) find F2. (7)

The acceleration of P is (3i + 9j) ms-2. At time t = 0, the velocity of P is (3i - 22j) ms-1

(b) Find the speed of P when t = 3 seconds. (4)

GCE A level (Part 2) Mathematics ? Section 10 ? Assignment 10

5

A particle P of mass 2 kg is pushed up a line of greatest slope of a rough plane by a horizontal force of magnitude X newtons, as shown in the diagram above. The force acts in the vertical plane which contains P and a line of greatest slope of the plane. The plane is inclined to the horizontal at an angle a, wheretana=

The coefficient of friction between P and the plane is 0.5.

Given that the acceleration of P is 1.45 ms–2, find the value of X.

(10)

6

A uniform rod AB has a length 4a and weight W. A particle of weight kW, k 1, is attached to the rod at B. The rod rests in equilibrium against a fixed smooth horizontal peg. The end A of the rod is on rough horizontal ground, as shown in the figure above. The rod rests on the peg at C, where

AC = 3a, and makes an angle a with the ground, where tana=.

The peg is perpendicular to the vertical plane containing AB.

(a) Give a reason why the force acting on the rod at C is perpendicular

to the rod. (1)

(b) Show that the magnitude of the force acting on the rod at C is

10 ( )

W 1+ 2k (4)

5

The coefficient of friction between the rod and the ground is .

(c) Show that for the rod to remain in equilibrium k. (7)

4 © 2018 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 10 ? Assignment 10

7 (In this question the unit vectors i and j are in a vertical plane, i being horizontal and j being vertically upwards.)

At t = 0 a particle P is projected from a fixed point O with velocity (7i +7v3j) ms-1. The particle moves freely under gravity. The position vector of a point on the path of P is (xi + yj) m relative to O.

g 2. (5)

(a) Show that y = 3x- x

98

(b) Find the direction of motion of P when it passes through the point

on the path where x = 20. (4)

At time T seconds P passes through the point with position vector (2?i + ?j) m where ? is a positive constant.

(c) Find the value of T. (4)

Total marks for Assignment 10 = 75

Submit your assignment

When you have completed your assignment, submit it to your tutor for marking. You may need to scan your work, graphs and diagrams so that they are in a digital format, and save your document as a pdf file. Your tutor will send you helpful feedback and advice to help you progress through the course.

Assignment 10

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 10.

Links to the assessment requirements

The assignment uses exam-type questions and will test your understanding of the concepts you have explored in this section. These address the following in Statistics and Mechanics part of the specification:

6.1

7.3, 7.4, 7.5

8.2, 8.4, 8.5, 8.6

9.1

How your tutor will mark your work

Your tutor will:

? check that you have answered each question ? check your workings as well as your answer ? give you feedback ? make suggestions about how you may be able to improve your work.

Are you ready to do the assignment?

Before you do the assignment, work through the topics in Section 10, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the assignment.

GCE A level (Part 2) Mathematics ? Section 10 ? Assignment 10

What to do

Answer as many questions as you can.

It is always important to show your working in your responses.

Guide time

The guide time for this assignment is 1 hour 30 minutes.

1

(a) As shown in the diagram above, forces 10 N and 5 N act along OA and OB respectively, where angle AOB = 60°.

The resultant of these two forces makes an angle of ?° with OA.

Find the value of ?. (3)

(b) The resultant of the three forces shown in the diagram above has a magnitude of R N and acts along OX. Find the values of

(i) P (ii) R (5)

2 © 2018 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 10 ? Assignment 10

2 A non-uniform plank AB has length 6 m and mass 30 kg. The plank rests in equilibrium in a horizontal position on supports at the points S and T of the plank where AS = 0.5 m and TB = 2 m.

When a block of mass M kg is placed on the plank at A, the plank remains horizontal and in equilibrium and the plank is on the point of tilting about S.

When the block is moved to B, the plank remains horizontal and in equilibrium and the plank is on the point of tilting about T.

The distance of the centre of mass of the plank from A is d metres. The block is modelled as a particle and the plank is modelled as a non-uniform rod. Find:

(a) the value of d

(b) the value of M. (7)

3 A boat B is moving with constant velocity. At noon, B is at the point with position vector (3i - 4j) km with respect to a fixed origin O. At 1430 on the same day, B is at the point with position vector (8i +11j) km. (a) Find the velocity of B, giving your answer in the form pi+qj. (3) At time t hours after noon, the position vector of B is b km.

(b) Find, in terms of t, an expression for b. (3)

Another boat C is also moving with constant velocity. The position vector of C, c km, at time t hours after noon, is given by c = (-9i + 20j) +t(6i + ?j), where ? is a constant. Given that C intercepts B,

(c) find the value of ? (5)

(d) show that, before C intercepts B, the boats are moving with the same speed. (3)

4 Two forces F1 and F2 act on a particle P.

The force F1 is given by F1 = (-i + 2j) N and F2 acts in the direction of the vector (i + j).

Given that the resultant of F1 and F2 acts in the direction of the vector (i + 3j),

(a) find F2. (7)

The acceleration of P is (3i + 9j) ms-2. At time t = 0, the velocity of P is (3i - 22j) ms-1

(b) Find the speed of P when t = 3 seconds. (4)

GCE A level (Part 2) Mathematics ? Section 10 ? Assignment 10

5

A particle P of mass 2 kg is pushed up a line of greatest slope of a rough plane by a horizontal force of magnitude X newtons, as shown in the diagram above. The force acts in the vertical plane which contains P and a line of greatest slope of the plane. The plane is inclined to the horizontal at an angle a, wheretana=

The coefficient of friction between P and the plane is 0.5.

Given that the acceleration of P is 1.45 ms–2, find the value of X.

(10)

6

A uniform rod AB has a length 4a and weight W. A particle of weight kW, k 1, is attached to the rod at B. The rod rests in equilibrium against a fixed smooth horizontal peg. The end A of the rod is on rough horizontal ground, as shown in the figure above. The rod rests on the peg at C, where

AC = 3a, and makes an angle a with the ground, where tana=.

The peg is perpendicular to the vertical plane containing AB.

(a) Give a reason why the force acting on the rod at C is perpendicular

to the rod. (1)

(b) Show that the magnitude of the force acting on the rod at C is

10 ( )

W 1+ 2k (4)

5

The coefficient of friction between the rod and the ground is .

(c) Show that for the rod to remain in equilibrium k. (7)

4 © 2018 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 10 ? Assignment 10

7 (In this question the unit vectors i and j are in a vertical plane, i being horizontal and j being vertically upwards.)

At t = 0 a particle P is projected from a fixed point O with velocity (7i +7v3j) ms-1. The particle moves freely under gravity. The position vector of a point on the path of P is (xi + yj) m relative to O.

g 2. (5)

(a) Show that y = 3x- x

98

(b) Find the direction of motion of P when it passes through the point

on the path where x = 20. (4)

At time T seconds P passes through the point with position vector (2?i + ?j) m where ? is a positive constant.

(c) Find the value of T. (4)

Total marks for Assignment 10 = 75

Submit your assignment

When you have completed your assignment, submit it to your tutor for marking. You may need to scan your work, graphs and diagrams so that they are in a digital format, and save your document as a pdf file. Your tutor will send you helpful feedback and advice to help you progress through the course.

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