NANYANG TECHNOLOGICAL UNIVERSITY

First Year Engineering Course

FE1073: An Introduction to Engineering and Practices

Laboratory Manual

For

Experiment C1

Resultants and Equilibrium of Forces

Laboratory: Protective Engineering

Location: N1.1 – B5 – 01

School of Civil andEnvironmental Engineering

[CEE]

Session 2013 / 2014

FE1073-C1

RESULTANTS AND EQUILIBRIUM OF FORCES

This experiment consists of two parts. In Part I, we will measure the force vectors and resultants and observe equilibrium of concurrent force systems. In Part II, we will measure the forces on an elastic body and check the equilibrium of the forces.

Part I: Resultants and equilibrium of concurrent forces

I.1. OBJECTIVE

To measure force vectors, force resultants, and observe equilibrium of concurrent force systems.

I.2. THEORY

In Figure 1, the object A is subjected to 2 forces indicated by vectors F1 and F2. Since these forces are acting on the same point of the object A, they are called concurrent forces. As with any vector quantity, each force is defined both by its direction (the direction of the arrow) and by its magnitude, which is proportional to the length of the arrow.

Figure 1 Resultant of concurrent forces and equilibrant

The total force on the object A can be determined by adding vectors F1 and F2. In the illustration shown in Figure 1, the parallelogram method is used. The diagonal of the parallelogram defined by F1 and F2 is Fr, which is the vector indicating the magnitude and direction of the total force acting on the object. Fr is called the resultant of F 1 and F 2.

Another useful vector is Fe, the equilibrant of F1 and F2. Fe is the force needed to exactly offset the combined effect of F1 and F 2. F e has the same magnitude as Fr, but is in the opposite direction. As we will see in the following experiment, the equilibrant provides a useful experimental method for finding the resultant of two or more forces.

1

2

A

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I.3. EQUIPMENT

Experiment Board, Spring Balance, Degree Scale, Force Ring, Mass Hangers, Pulleys, Masses, String.

I.4. THE EXPERIMENT

Figure 2 Experimental setup for Part I

(1) Follow Figure 2 and use pulleys and hanging masses to setup the equipment so that two known forces, F1 ( ) and F2 ( ), are pulling the on the force ring. Use the Holding Pin to prevent the ring from moving. The Holding Pin provides a force, Fe, which is exactly opposite to the resultant of F1 and F2.

(2) Adjust the Spring Balance to determine the magnitude of Fe. As shown, keep the Spring Balance vertical and use a pulley to direct the force from the spring toward the desired direction. Move the Spring Balance toward or away from the pulley to vary the magnitude of the force. Adjust the pulley and Spring Balance so that the Holding Pin is centered in the Force Ring.

Note: to minimize the effects of the friction in the pulleys, tap as needed on the Experiment Board each time you reposition any component. This will help the Force Ring come to its true equilibrium position.

(3) Record the value of the hanging masses M1 and M2 (including the mass of the mass hangers); the magnitude in Newton of F1, F2 and Fe; and also 1, 2 and e, the angle each vector makes with respect to the zero-degree li ne on the degree scale.

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(4) Vary the magnitudes and directions of F1 and F2 and repeat steps (1) to (3) for one more time.

I.5. REPORT PART-I

(Accuracy up to 1 degree for angles and 0.01N for forces are required)

(i) Complete the data sheet and attach it to the report.

(ii) Use the values you recorded to construct F1, F2 and Fe on a separate sheet of paper. Choose an appropriate scale (such as 10cm/Newton) and make the length of each vector proportional to the magnitude of the force. Label each vector and indicate the magnitude of the force it represents.

(iii) On your diagram, use the parallelogram method to draw the resultant of F1 and F2. Label the resultant Fr. Measure the length of Fr to determine the magnitude of the resultant force and record this magnitude on your diagram.

(iv) Does the equilibrant force vector, Fe, exactly balance the resultant vector, Fr? If not, what is the percentage error? Discuss the possible sources of error in your measurements and constructions.

Further discussion for Formal Report

(v) For the given masses M1, M2 and the measured angles 1, 2, calculate the equilibrant Fe and its direction e using the equilibrium conditions. Compare the calculated results with the measured values and discuss possible ways to improve the measurement accuracy.

Part II: Equilibrium of rigid body

II.1. OBJECTIVE

To measure force vectors, arm of the force, and calculate all the moment of the rotational axis to check equilibrium of rigid body.

II.2. THEORY

In Figure 3a, an object is fixed at point A using a bolt such that it is free to rotate about A. If the object is subjected to a force F1 with an arm of L1 to point A, it will rotate about the bolt since there is a moment along the bolt with magnitude M (M=F1L1). However, it does not have any translation because the reaction at the bolt cancels out the force F1.

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(a) (b)

Figure 3(a) Rotating object subjected to one force and (b) Stable objective with two more forces.

In Figure 3b, the object is kept in balance when subjected to two more forces F2 and F3. The balance condition indicates that the sum of all the moments due to the three forces about the bolt must be zero, that is,

(1)

Please bear in mind that the object has no translation because the reaction at the bolt cancels out the three acting forces. This means that the sum of all forces in any directions must be zero. It is sufficient to consider only x and y directions for planar forces.

Consequently, we can conclude that the conditions of the equilibrium of a rigid body are:

(2)

Figure 4 Force model of the elastic model in the experiment

Figure 4 gives the free body diagram of the rigid arm used in the experiment. The rigid arm is fixed at point O’ but can rotate about O’. In this figure, F represents the total weight of hanging masses and mass hanger, T is the

L1 L2

L3

F1

F2

F3

X

Y

L1

F1

X

Y

AA

F

L1 L2

O'

mgT

h0

T cos θ

T sin θ

Fo'y

Fo'x

S

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reading from spring scale, mg is the weight of spring scales, FO’x and FO’y are the reactions at point O’ and θ is the angle between the spring scale and the horizontal direction. When the object is in balance, we can obtain three equations based on Equation (2). Since we are not interested in the reactions at point O’, we only apply the third equation to check the equilibrium. Taking moment about O’, the equilibrium equation can be expressed as follows:

F x L1 + (T cos ho = mg (L2 + S cos ) + (T sin L2 (3)

The left hand side contains the counter clockwise terms of the moment, and the right hand side contains the clockwise terms.

II.3. EQUIPMENT

Experiment packet, Drop Weight, Mass Hangers, Masses, Spring Scale, String, Rulers.

Materials Student needs to bring the following items:

a) 10cm Protractor (Marking angles)b) 30cm Rulerc) A4 Writing Padd) A4 graph papers (4 sheets)

II.4. EXPERIMENT

Figure 5 Experiment instrument

(1) As shown in Figure 5, the experiment instrument is fixed on the board. Use two drop weights, shown as arrowhead, to obtain two points A and O. Then adjust the instrument and make sure AO and the slip band are aligned. The point O should coincide with the left edge of the slip band. Measure the height of the instrument from O’ to the board and write down the value H. Measure the interval distance of two holes d0 and the height of the top of the instrument h0.

OA

O'

H

h0

d0

Slip band

Control roll

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Figure 6 Experimental setup for Part II

(2) Put two masses with the mass hanger at the 2nd hole on left side. Put the spring scale at the 2nd hole on the right side and make the string attached to and go through the control rolls.

(3) Using equation , calculate the value of L with a given angle (around

450). Measure a distance L from the edge of slip band O to point B. Move the left control to point B and fixed. Adjust the right control until the instrument is in balance (the rigid arm at top should be kept horizontal), as shown in Figure 6. Record the force T from the reading of spring scale.

(4) Vary the direction of the spring scale such that is around 600 and repeat step (3). Vary the direction of the spring scale again such that is around 750 and repeat step (3).

Note: The angle of the spring scale should not be too small because the maximum value of the spring scaling is 10N. In this experiment, the smallest angle of 45 degree is recommended.

(5) Change the masses such that the total weight F is 1.7 N. Move the mass hanger to the 1st hole on the left side and put the spring scale at the 2nd hole on the right side. Repeat steps (3) and (4).

(6) Change the masses such that the total weight F is 3.2 N. Move the mass hanger to the 2nd hole on the left side and put the spring scale at the 1st hole on the right side repeat steps (3) and (4). Please note that L is the distance from the edge of slip

band to the target position. ( )

B

O'

O L

h0

L2L1

H

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II.5. REPORT PART-II

(i) Complete the data sheet and attach it to the report.

(ii) Based on the theory mentioned above, use the values you recorded to calculate the counter clockwise (M+) and clockwise (M-) moments.

(iii) Does the counter clockwise moment M+ exactly balance the clockwise moment M-? If not, what is the percentage error? Discuss the possible sources of error in your measurements and calculations.

Further discussion for Formal Report

(iv) Shown as Figure 5, given the weights F with known position (2nd hole at the left), the angle of spring scale and the position of the spring scale (2nd hole at the right), predict the theoretical force T (ignore the weight of the spring scale).

(v) Discuss possible ways to improve the accuracies of the measurements.

Conclusions

Keeping the objectives of the experiment in mind, conclude your findings.

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Experiment C1: Resultants and Equilibrium of Forces

DATA SHEET

Name: Date:

Group No.: Lecturer:

Table 1: Resultant and equilibrium of concurrent forces

Case 1

M1

(g)M2

(g)1

(degree)2

(degree)F1=M1g

(N)F2=M2g

(N)Fe

(N)e

(degree)

Case 2

M1

(g)M2

(g)1

(degree)2

(degree)F1=M1g

(N)F2=M2g

(N)Fe

(N)e

(degree)

Table 2: Equilibrium of Rigid Body

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DATA SHEET

Case 1

F = 2.2N, Weight of spring scale mg = 0.48N.

F position: 2nd hole (left), T position: 2nd hole (right)

H (mm) d0 (mm) h0 (mm) L1 (mm) L2 (mm) S (mm)

Angleθ

DistanceL (mm)

RecordT (N)

Counter clockwiseMoment M+

(Nmm)

ClockwiseMoment M-

(Nmm)

Error(M+) – (M-) / (M+)

(%)

Around 45°

Around 60°

Around 75°

Involved equations:

M+: M-: mg (L2 + S cos ) + (T sin L2

[DO NOT show any calculations on the data sheets. Show the calculations neatly on separate sheets.]

Table 2: Equilibrium of Rigid Body

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DATA SHEET

Case 2

F = 1.7N, Weight of spring scale mg = 0.48N.

F position: 1st hole (left), T position: 2nd hole (right)

H (mm) d0 (mm) h0 (mm) L1 (mm) L2 (mm) S (mm)

Angleθ

DistanceL (mm)

RecordT (N)

Counter clockwise

Moment M+(Nmm)

ClockwiseMoment M-

(Nmm)

Error(M+) – (M-) / (M+)

(%)

Around 45°

Around 60°

Around 75°

Involved equations:

M+: M-: mg (L2 + S cos ) + (T sin L2

Table 2: Equilibrium of Rigid Body

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DATA SHEET

Case 3

F = 3.2 N, Weight of spring scale mg = 0.48N.

F position: 2nd hole (left), T position: 1st hole (right)

H (mm) d0 (mm) h0 (mm) L1 (mm) L2 (mm) S (mm)

Angleθ

DistanceL (mm)

RecordT (N)

Counter clockwiseMoment M+

(Nmm)

ClockwiseMoment M-

(Nmm)

Error(M+) – (M-) / (M+)

(%)

Around 45°

Around 60°

Around 75°

Involved equations:

M+: M-: mg (L2 + S cos ) + (T sin L2

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