The main aim of the experiment was to verify the vector cross product and its torque.
Other aims were to understand the mathematical model of carrying out the vector cross product and to determine the equilibrium in a system under torque.
In the manipulation of vectors through multiplication two kind of products are achieved, the dot product and the cross product. The dot product of a vector is a scalar quantity and hence it only has magnitude. An example of scalar quantity is work done by a force.
The cross product of a vector produces a quantity that has both direction and magnitude. The product of these vectors is another vector which lies perpendicular to the plane of the primary vectors. The primary use of the cross product is to determine torque produced by a certain force acting at a distance onto an object.
In the multiplication of the cross product, for example if we use vectors A and B, then the product is given as C as shown below.
The vector C can be obtained by multiplying the magnitudes of the vectors A and B and by the sine of the angle between vectors A and B.
The area formed by the triangle between vectors A and B is equal to the magnitude of vector C. The direction of this vector is perpendicular to the plane of the two composing vectors.
The derivation of the vector cross product to form another vector perpendicular to its root vectors owes it origins from naturally occurring phenomena such as the correlation in the electrical world of the force produced in a motor in relation to the current and inductance in the coil. The relation is given as shown below.
Where I represents the electric current in amperes, L the length of the given wire and B the magnetic field strength.
Forces that are found to have a turning effect on an object are said to produce a torque. Where the torque is the rotational effect due to a force. This effect is usually experienced in many day today activities such as pistons which produces the torque required to drive a vehicle. The main source of the force is from the expanding gases during the power stroke of the combustion cycle. Another relative term is the bending moment that is also related to the torque but it is produced in relatively the same way.
Torque is usually represented using the term τ.
If we consider a hexagonal nut being tightened by a spanner as shown in the figure below
If the force being used to tighten the nut is parallel in the direction of y shown then the turning effect that is used to tighten or loosen the nut is given by the vector cross product of vector F and vector R. Where vector F is the tightening force and Vector R is the position vector of the point of action of the force from the reference position ie the origin.
Therefore the turning effect can be given as
Where the τ is the torque being applied and γ is the angle between the force applied and the direction vector.
The magnitude torque produced can be found using two ways. The first method utilizes the perpendicular distance to the force vector. Then the perpendicular distance is multiplied with the magnitude of the force vector thereby producing the torque. The equation is shown below.
The other method involves investigating the force component that is perpendicular to the position vector. Therefore the vector cross product magnitude gives the following solution;
However, while using the two methods described above, it is important to note the direction of rotation by using the right hand rule. Which dictates that if the force points in the direction of the fingers of the clenched right hand the torque produced is in the direction pointed by the thumb rotating in the direction pointed by the other fingers. This is important since some forces will produce positive moments which are usually in the anticlockwise direction and others will produce anticlockwise moments in opposite direction and thus they may tend to counteract and therefore it is needed to find the overall moment produced.
Also the vector cross product can be found in a mathematical manner by multiplying the forces as shown below. First the forces are represented in their components in x, y and z directions.
Then the cross product represented by vector C is found by finding the determinant of the matrix shown below.
The mathematical model is sometimes preferred by some users since there is no need to use the right hand rule. These three methods are used to determine the torque produced by forces about a certain point.
In order to justify the vector cross product application, a torque wheel is used.
The moment arm represents the shortest perpendicular distance from the line of action of force to the pivot or center of rotation. Forces are applied at various location in the holes. The torque produced by each force can be calculated by any of the method described earlier.
- Torque wheel
- Mounting board
- Weight hangers
- Clear plastic bars
The Excel file cross product calculator was opened and suitable equations were programmed onto the template to work out the x, y and z components of the cross product of two vectors provided. Then using vector algebra, the angle between vectors A and B was calculated. Then using the mathematical approach of finding the cress product, then all the components of vector c were programmed onto respective cells. The equation was then tested.
Using vector algebra the magnitudes of vectors A, B and C were calculated on row4
The torque wheel was mounted on the board. The three forces were attached on the torque wheel using the clear plastic bars. The plastic bars were pivoted on a small pin that was pinned on the board. The strings were then strung over the pulleys and the ends attached to weight hangers. Then weights were added to the weight hangers until there was an appreciable weight on each hanger. Care was taken to avoid equal torques that would cancel each other thus having zero torque at one end.
The equilibrium of the system was tested by rotating the wheel a small distance and if the system returned to original position then the equilibrium was assumed to be achieved.
The moment arms were measure for each force and the torque calculated. The torques were added and then then if the sum was not zero the remainder was noted to be the residual torque. Then the relative proportion between the residual and the average torque was worked out as a percentage and recorded.
The errors that were encountered were then analyzed to find their sources. Theories of sources of errors were postulated and examined to find out if they exhibited any correlation between the theoretical and experimental errors that were encountered.
The mass of the rod along with its riders was measured then its center of mass found. Then a third rider was mounted on the rod center of mass. An assembly similar to the one given below was made.
Masses were added to both hangers and the largest one falling at the farthest hanger to represent a hand lifting a load. The weights were adjusted so that the angle between the bicep and the hand was right angle.
A net of Cartesian plane originating from the pivot pin was drawn. The lengths of each moment arm were measured and the error emanating from excess torque calculated. The x and y components of the forces in the bone were calculated.
Mass of rod = 38.6g
Error = ±0.1cm
Angle θ = 1.872 radians
R was = 30.6 mm
Error =0.12+0.121=0.1414 =
In the torque wheel experiment the torque was found by
Force = mass × gravity acceleration
0.055 × 9.81 = 0.5396 Newtons
Torque = Force × moment arm
0.5396 × 0.024 = 0.01295 N-m
The moment produced by the force C is counter clockwise hence negative.
Average torque used in the board = Total torque ÷3
(0.01295 + 0.0086 +0.0206) ÷ 3 = 0.01405
Percentage of residual torque to average torque = 0.00095 ÷ 0.01405 ×100
Theoretical error from the accuracy is calculated as
Error = ±0.1cm
In the Pasco mechanism systems experiment
The residual torque was found to be 0.00095N-m which constituted a percentage of about 6.76% in experimental error. The experimental error was found to 6.76% while the theoretical error was 14.14%. The disparity in the errors can pegged to the friction in the bearing that was used as the pivot point in the board.
The length of the moment arm for the force exerted by the pin is zero since it exerts on the pivot.
The advantage of having small displacements at the biceps is that one does not need very long biceps but small compact biceps the allow for very small displacements
The aim of the project was to investigate the use of the vector cross product to in finding torque and moments in apparatus. The background provided a rich ground from whom the method was derived. The procedure was followed and the right values of results were achieved the led to success in fulfilling of the prime objectives. Hence the application of the vector cross product was utilized in torque wheel and human arm model.