What Hurts the Most
Content
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
EQUILIBRIUM OF FORCES IN THREE-DIMENSIONS OBJECTIVES 1. To investigate the equilibrium of forces in three-dimensions. 2. To prove the summation of forces in x, y and z directions are zeros Apparatus: 1. A set of 3D force apparatus (force table with protractor and pulleys) 2. A roll of cord (string) 3. A set of weights
1
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
How does this work: y y y y y y y y In this experiment, we are going to use the force table shown in Figure 1 and 2. Three set of weights (W1, W2 and W3) are connected to cords and hung to 3 separate poles with pulley installed on them. The three cords are connected to a ring that will be inserted to the centre pole. The three poles can be adjusted to be at any point along the table s circumference. The weights can be added and taken away until the ring is on the centre of the centre pole. Then, from Figure 1, we are able to adjust the heights H1, H2, H3, measure H and the radius of the table R. From that we can calculate the z-component of the forces and also the x, y, components of the forces. From the x, y-components of the forces, we are able to calculate the x-component and the y-component of the forces after we measure the angle 1, 2 and 3.
Theory: y y y Referring to Figure 1 and 2, assuming that the pulley is frictionless, then the force in the string is equal to the weight hanging at its end. is the angle between the string and its projection on the xy plane. is the angle which the projection of the cord made with the x axis.
a. Force in the z -direction =
§W
i
sin E i = 0 =0 =0
b. Component of the force in the xy plane = Wi cos E i c. Force in the y direction = d. Force in the x
§ (W cos E ) sin U direction = § (W cos E ) cosU
i i i i
i i
2
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
EXPERIMENTAL PROCEDURE: 1. 2. 3. 4. 5. 6. The location of the cord is first selected. Then the vertical poles are clamped at the selected locations. The height of the pulley is then adjusted and locked in its position. One end of the cord is tied to the ring while the ring is slipped into central. Each cord is passed over its respective pulley where a load hanger is attached to the other end of the cord. The position of the central ring is then checked. (The Centre Pole should be aligned at the centre of the ring, if not; this position should be adjusted properly by loading appropriate weight on to the load hanger). Desired weight is placed on two of the load hangers. Then, gradually weight is added onto the third hanger so that the centre pole is aligned at the centre of the ring without touching the ring in anyways. Finally the weights on each hanger and the location of the cord are measured accordingly. This experiment is then repeated several times using different sets of locations loading appropriate sets of weight.
7. 8. 9. 10.
ACQUIRED DATA Contains data obtained from the experiment after four runs.
Pulley 1 W1 N 0.1 0.5 0.5 0.9 H1 cm 32.0 30.2 30.0 30.0
1
Pulley 2 W2 N 0.5 0.5 0.2 0.5 H2 cm 30.0 31.5 31.2 31.4
2
Pulley 3 W3 N 0.8 1.7 0.7 1.2 H3 cm 18.2 18.5 24.0 30.0
3
H(cm)
degree 60 60 20 20
degree 120 120 110 110
degree 280 280 240 240 29.5 30.5 30.0 30.0
3
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
METHODS OF CALCULATION:
4
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
RESULTS The following results can be calculated using the data obtained in Table 1. (a) Component of force in the z direction, Fz (positive upwards)
Length of cord, L =
(H 1 H 2 ) 2 R 2
sin
1
=
H1 H L
1
Fz = W1 sin
Pulley 1 W1 N 0.1 0.5 0.5 0.9 sin
1
Pulley 2 Fz1 N 0.0111 -0.0067 0 0 W2 N 0.5 0.5 0.2 0.5 sin
2
Pulley 3 Fz2 N 0.0527 0.0224 0.0107 0.0314 W3 N 0.8 1.7 0.7 1.2 sin
3
Fz3 N 0 -0.0153 0 0.0108
Fz N 0.0638 0.0004 00107 0.0422
0.1114 -0.0134 0 0
0.1053 0.0448 0.0537 0.0627
0 0.0090 0 0.0090
Table 2- Equilibrium of forces in the z-direction
5
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
From Table 2 above, calculate the total force in the z-direction, FzT, for each load case
FzT ! § Fz W1 W2 W3
§ F [N ]
z
FZT -1.33 -2.69 -1.38 -2.55
0.0638 0.0004 00107 0.0422
b) Component of the force in the xy plane, Fxy = W cos Component of the force in the x direction, Fx = W cos
F(xy) = W cos cos
Pulley 1 W1 N 0.1 0.5 0.5 0.9 cos 0.5 0.5 0.94 0.94
1
Pulley 2
1
Pulley 3
2
cos
0.9938 0.9999 1.0000 1.0000
Fx1 N 0.0497 0.2500 0.4700 0.8460
W2 N 0.5 0.5 0.2 0.5
cos -0.5 -0.5
2
cos
0.9944 0.9990 0.9986 0.9980
Fx2 N -0.2486 -0.2498 -0.0679 -01697
W3 N 0.8 1.7 0.7 1.2
cos
3
cos
3
0.17 0.17 -0.5 -0.5
1.0000 1.0000 1.0000 1.0000
Fx3 N 0.1360 0.2890 -0.3500 -0.6000
Fx N -0.0629 0.2892 0.0521 0.0763
-0.34 -0.34
Table 3- Equilibrium of forces in the x- direction From Table 3, calculate the total force in the x-direction, Fxt, for each load case,
§F
xT
! § Fx
FxT -0.0629 0.2892 0.0521 0.0763
Fx -0.0629 0.2892 0.0521 0.0763
6
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
(c) Component of the force in the xy plane, Fxy = W cos Component of force in the y direction, Fy = W cos sin
Pulley 1 W1 N 0.1 0.5 0.5 0.9 sin
1
Pulley 2 Fy1 W2 N 0.5 0.5 0.2 0.5 sin
2
Pulley 3 Fy2 W3 N 0.8 1.7 0.7 1.2 sin
2
Fy3 cos
2
Fy N -0.2649 -0.7964 -0.2513 -0.2689
cos
1
N
0.0865 0.4350 0.1700 0.3060
cos
2
N 0.4326 0.4346 0.1877 0.4691
N -0.7840 -1.6660 -0.6090 -1.0440
0.87 0.87 0.34 0.34
0.9938 0.9999 1.0000 1.0000
0.87 0.87 0.94 0.94
0.9944 0.9990 0.9986 0.9980
-0.98 -0.98 -0.87 -0.87
1.0000 1.0000 1.0000 1.0000
Table 4- Equilibrium of forces in the y direction
Summation of the forces and divided by the mean Force in the z-direction = 0.0072 + 0.0179 + 0.0179 + 0.0027 = 0.0457 (Approximately equal to zero) Component of the force in the x-y plane = 0.8 x 0.999 = 0.7992 Force in the y-direction = 0.338 + 0.433 + 0.787 + 0.064 = 1.622 (Very close to zero/Negligible) Force in the x-direction = -0.724 + -0.250 + 0.139 + -0.077 = -0.912 (Very close to zero/Negligible) Hence the body remains in equilibrium in three dimensions.
7
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125
Equilibrium Of Forces In Three-Dimensions (Lab Report)
DISCUSSION
This experiment can practically produce values which are very close to the theoretical exact values. The pure eye has to see the body hanging in equilibrium to record the values and therefore the values come very close to the expected true values.
Still, there are certain experimental errors which are faced due to various reasons. Some of which are:
1. The body is considered to be in equilibrium when the centre pole stays through the ring without touching it, but still not sure whether the centre pole goes right through the centre of the ring. 2. The wind and other environmental forces can affect the weight when measuring because these forces can add in as extra weight whereas the exact load will be read as a lesser value. 3. The heights were measured using a feet scale without considering too much on its decimal values which can still cause changes in final calculations. 4. The summation of the three forces is supposed to be zero according to the theory. 5. Parallax error will affect the measurement. Therefore, proper position to take the measurement is needed. 6. There is friction along the three pulleys that cause errors in the experiment. 7. The values obtained from the measurement are not that accurate because the apparatus used, like the meter ruler have a lower accuracy and precision. 8. If we compare each and every reading with other group members we can see our readings and their readings are difference due to tolerance.
µ=± 0.3 (this value depends in each and every scenarios )
Therefore, the calculations of this experiment could be not exact expectations but still are approximately true values. I personally don t agree the way did this experiment which was six person per group, because everyone don t get a chance and everyone will not get a equal good knowledge, I recommend this experiment should be done max three person and minimum two people.
8
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
SUMMARY
Through this experiment, the concept of a body remaining in equilibrium under three dimensional forces can be proved precisely with no doubt. There are several factors like the friction and parallax error had affected the calculation. If the errors can be prevented, the reading will be more accurate. This ends up with enough and more facts and calculations to show the summations of all three dimensional forces acting on a body when it is in equilibrium and hence good enough to come up with a worthy discussion. The summation of each three dimensional force can be proved zero (If the centre pole lies through the exact centre of the ring). Still with experimental errors distracting in between, the summation of each three dimensional force can be proved as a very close value to zero Newton s, where the values can even be neglected. In conclusion, this experiment succeeds in proving concepts on equilibrium of forces acting in three dimensions.
Conclusion Theoretically, the summation of forces should be zero because it is in equilibrium. However, several factor like the friction and parallax error had affected the calculation. If the errors can be prevented, the reading will be more accurate.
9
Bradly Roger Shedden (7433670)
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
EQUILIBRIUM OF FORCES IN THREE-DIMENSIONS OBJECTIVES 1. To investigate the equilibrium of forces in three-dimensions. 2. To prove the summation of forces in x, y and z directions are zeros Apparatus: 1. A set of 3D force apparatus (force table with protractor and pulleys) 2. A roll of cord (string) 3. A set of weights
1
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
How does this work: y y y y y y y y In this experiment, we are going to use the force table shown in Figure 1 and 2. Three set of weights (W1, W2 and W3) are connected to cords and hung to 3 separate poles with pulley installed on them. The three cords are connected to a ring that will be inserted to the centre pole. The three poles can be adjusted to be at any point along the table s circumference. The weights can be added and taken away until the ring is on the centre of the centre pole. Then, from Figure 1, we are able to adjust the heights H1, H2, H3, measure H and the radius of the table R. From that we can calculate the z-component of the forces and also the x, y, components of the forces. From the x, y-components of the forces, we are able to calculate the x-component and the y-component of the forces after we measure the angle 1, 2 and 3.
Theory: y y y Referring to Figure 1 and 2, assuming that the pulley is frictionless, then the force in the string is equal to the weight hanging at its end. is the angle between the string and its projection on the xy plane. is the angle which the projection of the cord made with the x axis.
a. Force in the z -direction =
§W
i
sin E i = 0 =0 =0
b. Component of the force in the xy plane = Wi cos E i c. Force in the y direction = d. Force in the x
§ (W cos E ) sin U direction = § (W cos E ) cosU
i i i i
i i
2
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
EXPERIMENTAL PROCEDURE: 1. 2. 3. 4. 5. 6. The location of the cord is first selected. Then the vertical poles are clamped at the selected locations. The height of the pulley is then adjusted and locked in its position. One end of the cord is tied to the ring while the ring is slipped into central. Each cord is passed over its respective pulley where a load hanger is attached to the other end of the cord. The position of the central ring is then checked. (The Centre Pole should be aligned at the centre of the ring, if not; this position should be adjusted properly by loading appropriate weight on to the load hanger). Desired weight is placed on two of the load hangers. Then, gradually weight is added onto the third hanger so that the centre pole is aligned at the centre of the ring without touching the ring in anyways. Finally the weights on each hanger and the location of the cord are measured accordingly. This experiment is then repeated several times using different sets of locations loading appropriate sets of weight.
7. 8. 9. 10.
ACQUIRED DATA Contains data obtained from the experiment after four runs.
Pulley 1 W1 N 0.1 0.5 0.5 0.9 H1 cm 32.0 30.2 30.0 30.0
1
Pulley 2 W2 N 0.5 0.5 0.2 0.5 H2 cm 30.0 31.5 31.2 31.4
2
Pulley 3 W3 N 0.8 1.7 0.7 1.2 H3 cm 18.2 18.5 24.0 30.0
3
H(cm)
degree 60 60 20 20
degree 120 120 110 110
degree 280 280 240 240 29.5 30.5 30.0 30.0
3
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
METHODS OF CALCULATION:
4
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
RESULTS The following results can be calculated using the data obtained in Table 1. (a) Component of force in the z direction, Fz (positive upwards)
Length of cord, L =
(H 1 H 2 ) 2 R 2
sin
1
=
H1 H L
1
Fz = W1 sin
Pulley 1 W1 N 0.1 0.5 0.5 0.9 sin
1
Pulley 2 Fz1 N 0.0111 -0.0067 0 0 W2 N 0.5 0.5 0.2 0.5 sin
2
Pulley 3 Fz2 N 0.0527 0.0224 0.0107 0.0314 W3 N 0.8 1.7 0.7 1.2 sin
3
Fz3 N 0 -0.0153 0 0.0108
Fz N 0.0638 0.0004 00107 0.0422
0.1114 -0.0134 0 0
0.1053 0.0448 0.0537 0.0627
0 0.0090 0 0.0090
Table 2- Equilibrium of forces in the z-direction
5
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
From Table 2 above, calculate the total force in the z-direction, FzT, for each load case
FzT ! § Fz W1 W2 W3
§ F [N ]
z
FZT -1.33 -2.69 -1.38 -2.55
0.0638 0.0004 00107 0.0422
b) Component of the force in the xy plane, Fxy = W cos Component of the force in the x direction, Fx = W cos
F(xy) = W cos cos
Pulley 1 W1 N 0.1 0.5 0.5 0.9 cos 0.5 0.5 0.94 0.94
1
Pulley 2
1
Pulley 3
2
cos
0.9938 0.9999 1.0000 1.0000
Fx1 N 0.0497 0.2500 0.4700 0.8460
W2 N 0.5 0.5 0.2 0.5
cos -0.5 -0.5
2
cos
0.9944 0.9990 0.9986 0.9980
Fx2 N -0.2486 -0.2498 -0.0679 -01697
W3 N 0.8 1.7 0.7 1.2
cos
3
cos
3
0.17 0.17 -0.5 -0.5
1.0000 1.0000 1.0000 1.0000
Fx3 N 0.1360 0.2890 -0.3500 -0.6000
Fx N -0.0629 0.2892 0.0521 0.0763
-0.34 -0.34
Table 3- Equilibrium of forces in the x- direction From Table 3, calculate the total force in the x-direction, Fxt, for each load case,
§F
xT
! § Fx
FxT -0.0629 0.2892 0.0521 0.0763
Fx -0.0629 0.2892 0.0521 0.0763
6
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
(c) Component of the force in the xy plane, Fxy = W cos Component of force in the y direction, Fy = W cos sin
Pulley 1 W1 N 0.1 0.5 0.5 0.9 sin
1
Pulley 2 Fy1 W2 N 0.5 0.5 0.2 0.5 sin
2
Pulley 3 Fy2 W3 N 0.8 1.7 0.7 1.2 sin
2
Fy3 cos
2
Fy N -0.2649 -0.7964 -0.2513 -0.2689
cos
1
N
0.0865 0.4350 0.1700 0.3060
cos
2
N 0.4326 0.4346 0.1877 0.4691
N -0.7840 -1.6660 -0.6090 -1.0440
0.87 0.87 0.34 0.34
0.9938 0.9999 1.0000 1.0000
0.87 0.87 0.94 0.94
0.9944 0.9990 0.9986 0.9980
-0.98 -0.98 -0.87 -0.87
1.0000 1.0000 1.0000 1.0000
Table 4- Equilibrium of forces in the y direction
Summation of the forces and divided by the mean Force in the z-direction = 0.0072 + 0.0179 + 0.0179 + 0.0027 = 0.0457 (Approximately equal to zero) Component of the force in the x-y plane = 0.8 x 0.999 = 0.7992 Force in the y-direction = 0.338 + 0.433 + 0.787 + 0.064 = 1.622 (Very close to zero/Negligible) Force in the x-direction = -0.724 + -0.250 + 0.139 + -0.077 = -0.912 (Very close to zero/Negligible) Hence the body remains in equilibrium in three dimensions.
7
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125
Equilibrium Of Forces In Three-Dimensions (Lab Report)
DISCUSSION
This experiment can practically produce values which are very close to the theoretical exact values. The pure eye has to see the body hanging in equilibrium to record the values and therefore the values come very close to the expected true values.
Still, there are certain experimental errors which are faced due to various reasons. Some of which are:
1. The body is considered to be in equilibrium when the centre pole stays through the ring without touching it, but still not sure whether the centre pole goes right through the centre of the ring. 2. The wind and other environmental forces can affect the weight when measuring because these forces can add in as extra weight whereas the exact load will be read as a lesser value. 3. The heights were measured using a feet scale without considering too much on its decimal values which can still cause changes in final calculations. 4. The summation of the three forces is supposed to be zero according to the theory. 5. Parallax error will affect the measurement. Therefore, proper position to take the measurement is needed. 6. There is friction along the three pulleys that cause errors in the experiment. 7. The values obtained from the measurement are not that accurate because the apparatus used, like the meter ruler have a lower accuracy and precision. 8. If we compare each and every reading with other group members we can see our readings and their readings are difference due to tolerance.
µ=± 0.3 (this value depends in each and every scenarios )
Therefore, the calculations of this experiment could be not exact expectations but still are approximately true values. I personally don t agree the way did this experiment which was six person per group, because everyone don t get a chance and everyone will not get a equal good knowledge, I recommend this experiment should be done max three person and minimum two people.
8
Bradly Roger Shedden (7433670)
2011,Apr. 17
Mechanics of Structures HES1125 Equilibrium Of Forces In Three-Dimensions (Lab Report)
SUMMARY
Through this experiment, the concept of a body remaining in equilibrium under three dimensional forces can be proved precisely with no doubt. There are several factors like the friction and parallax error had affected the calculation. If the errors can be prevented, the reading will be more accurate. This ends up with enough and more facts and calculations to show the summations of all three dimensional forces acting on a body when it is in equilibrium and hence good enough to come up with a worthy discussion. The summation of each three dimensional force can be proved zero (If the centre pole lies through the exact centre of the ring). Still with experimental errors distracting in between, the summation of each three dimensional force can be proved as a very close value to zero Newton s, where the values can even be neglected. In conclusion, this experiment succeeds in proving concepts on equilibrium of forces acting in three dimensions.
Conclusion Theoretically, the summation of forces should be zero because it is in equilibrium. However, several factor like the friction and parallax error had affected the calculation. If the errors can be prevented, the reading will be more accurate.
9
Bradly Roger Shedden (7433670)
