New Zealand Diploma

in Engineering Level 6

MECH4001 (DE4101)

ENGINEERING FUNDAMENTALS

VERSION 2 – LEVEL 4 CREDIT 15

Laboratory 1

Concurrent Forces (Vectors)

GENERAL INSTRUCTIONS

• Enter your name above.

• Complete these exercises during scheduled class time.

• Have each exercise checked by the tutor when complete.

• Due Date:

EXPERIMENTS 1 and 2: TRIANGLE OF FORCES

OBJECTIVE

To test that three non-parallel forces in equilibrium can be represented by a triangle of forces, from which two of the forces can be found when the third force is known, provided that the direction or line of action of the forces is known.

INTRODUCTION

When three forces in the same plane act in different directions on a stationary body their lines of action meet at a point. Because of this the forces can be represented by a force diagram called the Triangle of Forces. This can be used to find the size of two of the forces when the third force is known. For example, when a weight W is supported by two ropes positioned as in Fig 1, the pull in the ropes, F1 and F2, can be found by drawing the triangle of forces.

Fig 1. Experiments 1 and 2 (Three Forces) with Free Body Diagram 

EQUIPMENT

• 1 Diagram board with clip

• 2 short screws

• 3 Pulleys

• 5 knurled nuts

• 3 Weight hooks

• 1 Set of weights

• 1 Ring with three cords attached

• Some sheets of graph paper

METHOD – Experiment 1

Fig 2. Experiments 1 and 2 (Three Forces) [Set-up]

• Position the diagram board (horizontal / landscape position) and secure with screws and nuts through holes H7 and H12. Clip a sheet of paper to the diagram board.

• Position the three pulleys in holes 5K, 13K and 11D.

• Pass two of the ring cords over the upper pulleys and attach weight hooks. Attach another weight hook to the third cord and let it hang directly from the ring. This should bring the ring near the centre of the diagram board.

• Hold the ring against the diagram board and add weights to give values of 2.7N, 3.2N and 2.2 Newton as shown in Fig 2. (Each weight hook weighs 0.1N)

• Release the ring and gently cause the system to “bounce” by jogging the CENTRE weight only and let it settle freely in its equilibrium position.

• Mark the positions of the three cords with pencil dots on the paper.

• Remove the paper, draw the lines representing the three cords, and write in the weight supported by each cord. 

METHOD – Experiment 2

Clip a new sheet of paper to the diagram board. Slacken the nuts which secure the diagram board and slide the diagram board to the right as far as the slots will allow and retighten the screws. Keep the weights the same as for test A, but this time pass the centre weight cord over the lower pulley as shown dotted in Fig 2. This will alter the angle between the cords. Once again, lightly “bounce” the centre weight and allow it to settle in its new equilibrium position.

Mark the new position of the three cords and draw the three lines representing the cords as before, recording the weight supported on each cord.

RESULTS

On each of your drawings of the cord positions, mark the space in between the forces with letters a, b and c. This will be the space diagram showing the space position of the forces (Space diagram, Fig 3).

Fig 3. Space Diagram (Free Body Diagram) and matching

Force Diagram (Vector Diagram) for three forces

Draw the force diagram in a convenient space on the same sheet. To do this start with the known weight W1 and draw a line parallel to the middle cord, and along it mark off a length ab to a suitable scale (e.g. 10mm = 0.05 N) to represent the force equal to the middle weight W1 (Force diagram, Fig 3). Through b, draw a line parallel to the direction of the cord supporting the weight W2. Through a, draw a third line parallel to the direction of the third cord, to meet the second line at c. Then abc is the force diagram, or triangle of forces, for the three forces W1, W2 and W3.

Measure the lengths of bc and ca. These lengths should be equivalent to the corresponding weights W2 and W3. Check this.

CONCLUSION (Experiments 1 & 2)

From your results, write down the method of drawing the triangle of forces for three forces in equilibrium, and say how two of the forces can be found when the third force is known. When doing this, keep the following points in mind:

1 What does a space diagram show?

2 Does a space diagram need a scale?

3 Why is a scale needed for a force diagram?

4 When drawing the force diagram, which line is drawn first?

5 What does the Triangle of Forces show in addition to the magnitude of the force?

6 If your force diagrams did not close fully, what are the likely causes of errors, and how could they be minimised?

7 How would you quantify the errors in your diagrams in terms of force and or angles?

8 Describe the concept of static equilibrium with regard to forces.

EXPERIMENTS 3 and 4: POLYGON OF FORCES

OBJECTIVE

To test that when four or more forces are in equilibrium at a point, they can be represented by a polygon of forces, from which unknown forces can be found.

INTRODUCTION

In the design of pin-jointed plane structures such as girders, bridges and roof trusses (Fig 4) it is necessary to find the forces acting in each member so that the frame can be made strong enough to withstand the maximum loads exerted upon it. The polygon of forces is frequently employed to find such forces and deals with each joint in turn. This experiment could be regarded as one such joint on a structure, and it will be shown that in a system containing four or more forces, two unknowns can be found in magnitude or direction if the remaining information is known. The polygon of forces is an extension of the triangle of forces, and whereas ‘Tri’ means 3, ‘Poly’ means many.

EQUIPMENT

• 1 Diagram board with clip

• 2 short screws

• 4 Pulleys

• 6 knurled nuts

• 5 Weight hooks

• 1 Set of weights

• 1 Ring with five cords attached

• Some sheets of graph paper

Fig 4. Experiments 3 and 4 (Four and Five Forces respectively) [Set-up]

METHOD – Experiment 3

• Secure the mounting panel in the vertical position (“portrait”). Mount the diagram board vertically and secure the screws and nuts through holes 8D and 8K. Mount the four pulleys as shown in holes 4I, 5E, 11D and 12J.

• Clip a sheet of paper to the board and assemble with cords and weight hooks as shown.

• Add weights to give total weights as shown in diagram (Fig 4), total weight includes weight hook of 0.1N.

• The fifth cord (only needed in test 2) can be allowed to hang freely and will not affect Test 1 results.

• Gently cause the system to “bounce” by jogging the free cord and letting it settle freely in its equilibrium position. Then mark the position of the four cords with pencil dots on the paper. Remove the paper, join up the dots representing the cords, and write in the weight supported by each cord.

RESULTS – Experiment 3

On your diagram sheet for Test 1, mark the spaces between the cord lines with the letters A, B, C and D to give the space diagram (Fig 5).

Fig 5. Space Diagram (Free Body Diagram) and matching

Force Diagram (Vector Diagram) for four forces

In a convenient space draw a Force diagram on the same sheet by first drawing scale lengths ab and bc to represent the forces W1 and W2. Then through c and a draw lines parallel to the directions of W3 and W4 to meet d. The figure abcd is the force diagram, or polygon of forces, for the four forces W1, W2, W3 and W4. Measure the lengths cd and da. These should be equivalent to the corresponding forces W3 and W4. Check this.

METHOD – Experiment 4

Keeping the weights and pulleys as in Test 1, attach a weight hanger to the fifth cord [W5] and let it hang directly from the ring with a total weight of 1.1 N (including hook). Once again “bounce” the system by jogging the centre weight only, and allow the ring to settle in its new equilibrium position, then mark the position of the five cords and write in the corresponding weights.

RESULTS – Experiment 4

Draw the space diagram A, B, C, D, E to show the space positions of the five forces, W1, W2, W3, W4 and W5 (Fig 6.)

Fig 6. Space Diagram (Free Body Diagram) for five forces

Draw a separate force diagram starting with scale lengths ab, bc and cd, to represent the forces W1, W2 and W3. Complete the diagram by drawing lines parallel to the directions of W4 and W5, to give the point e. The figure abcde is the force diagram, or polygon of forces, for the five weights W1, W2, W3, W4 and W5. Check that the lengths de and ea are equivalent to the corresponding forces W4 and W5.

CONCLUSION (Experiments 3 & 4)

From your results, write down the method of using the Polygon of Forces for four or more forces in equilibrium at a point, and say how many unknown forces can be found. Keep the following points in mind:

9 What must be known about all the forces before the space diagram can be drawn?

10 How many of the force lines were not marked off in the force diagram?

11 What do you notice about the arrows showing the direction of the forces in the force diagram?

12 Can we find the direction of forces by this method if all other data is given?