All questions of System of Forces for Civil Engineering (CE) Exam

Moment is the turning effect produced.
the moment is the force acting on the body and the perpendicular distance of a point and the line of action of the force.

The moment of the force about the axis of rotation by the application of the force on the body is given by the cross product of both. If the force not perpendicular to the axis, and making angle θ then cosine form of angle is used. As usually used in the cross product.

The net forces acting on a body must be zero for equilibrium

In order for a body to be in equilibrium, the net forces acting on it must be zero. This means that the vector sum of all the forces must be equal to zero. If there is a non-zero net force acting on a body, it will accelerate in the direction of that force.

Applicability of this principle to simply supported structures

This principle is also applicable to simply supported structures. Simply supported structures are those that are supported at two points, such as a beam supported on two pillars. The support reactions at these points play an important role in determining the equilibrium of the structure.

Support reactions are counted in making the net force zero

The support reactions at the points of support in a simply supported structure are considered external forces acting on the body. These support reactions are typically in the form of normal forces and moments.

When analyzing the equilibrium of a simply supported structure, the support reactions must be taken into account. These reactions contribute to the net force acting on the structure and must be included in the calculation of the net force.

For example, consider a simply supported beam with a point load applied at the center. The support reactions at the two ends of the beam will contribute to the net force acting on the beam. If the magnitudes and directions of these support reactions are properly determined, they can be added to the applied load to ensure that the net force acting on the beam is zero.

Conclusion

In conclusion, the net forces acting on a body must be zero for equilibrium. This principle applies to simply supported structures as well. The support reactions at the points of support in a simply supported structure are considered external forces and must be included in the calculation of the net force. By properly determining the magnitudes and directions of these support reactions, the net force acting on the structure can be made zero, ensuring equilibrium.

The moment axis is always perpendicular to the planes of the force and the distance of the axis and the point of action of the force on the body. This means that the moment is the cross product of the force and the distance between the axis and the point of action of the force.

Yes, we can express the force in the Cartesian form.

Explanation:

Force is a vector quantity, which means it has both magnitude and direction. In order to fully describe a force, we need to specify its magnitude and direction.

The Cartesian form is a way to express vectors using their components along the coordinate axes. In a Cartesian coordinate system, we have three mutually perpendicular axes - x, y, and z. Any vector in this coordinate system can be expressed in terms of its components along these axes.

To express a force in Cartesian form, we need to determine its components along the x, y, and z axes. This can be done using vector addition and trigonometry.

Once we know the components of the force along each axis, we can write it in Cartesian form as:

F = Fx i + Fy j + Fz k

Here, F is the force vector, Fx, Fy, and Fz are the components of the force along the x, y, and z axes, and i, j, and k are unit vectors along the respective axes.

The advantage of expressing a force in Cartesian form is that it allows us to easily perform vector operations such as addition, subtraction, and dot product. It also helps in solving problems involving force equilibrium and resolving forces into their components.

In conclusion, the statement that we can express the force in the Cartesian form is true. By determining the components of the force along the x, y, and z axes, we can fully describe the force vector using the Cartesian coordinate system.

If A and B are two vectors and θ is the angle between them, A.B represents the dot product of A and B. The dot product of two vectors is defined as:

A.B = |A| |B| cos(θ),

where |A| and |B| represent the magnitudes of vectors A and B, respectively.

The answer is option 'B' - Concurrent forces.

Concurrent forces are forces that act on the same point or concurrent point. These forces have different magnitudes and directions, but they have a common point of application. When concurrent forces act on a body, they have a resultant force and a resultant moment. The resultant force is the vector sum of all the forces acting on the body, while the resultant moment is the vector sum of the moments caused by each force.

Non-concurrent forces are forces that do not have a common point of application. They have different points of application and can act at different angles and distances from each other. Non-concurrent forces can be parallel or non-parallel.

Rotation and forces
When forces act on a body, they can cause both translational motion and rotational motion. Translational motion refers to the linear movement of the body, while rotational motion refers to the spinning or turning motion of the body around an axis.

Concurrent forces and rotation
Concurrent forces do not cause rotation because their lines of action pass through the same point or are concurrent. Since the forces act along the same line, they produce a net force but no net moment about that point. This means that the forces do not create a turning effect or torque on the body, resulting in no rotation.

Non-concurrent forces and rotation
On the other hand, non-concurrent forces can cause rotation. When non-concurrent forces act on a body, their lines of action do not pass through the same point. As a result, they create a net moment or torque about a particular axis, leading to rotation.

Conclusion
In conclusion, the correct answer is option 'B' - Concurrent forces. Concurrent forces do not cause rotation because their lines of action pass through the same point, resulting in no net moment or torque. Non-concurrent forces, on the other hand, can cause rotation as their lines of action do not pass through the same point, creating a net moment or torque.

Length and Direction of Force Vector R

The force vector R is a fundamental concept in physics and engineering. It represents a force acting on an object and has both a length and a specific direction. The correct answer is option 'A', which states that the force vector R has a length equal to the magnitude of the force and a specific direction.

Length of R:
The length of the force vector R represents the magnitude or strength of the force. It is a scalar quantity and is usually measured in units such as Newtons (N) or pounds (lb). The length of R can be determined by measuring the force using appropriate instruments like force gauges or load cells. For example, if a force of 10 N is acting on an object, the length of vector R would be 10 units.

Direction of R:
The direction of the force vector R indicates the orientation or line of action of the force. It is a vector quantity and can be represented using various methods such as angles, coordinate systems, or Cartesian coordinates. The direction of R can be determined by using reference points or coordinate systems. For example, if a force is acting vertically downwards, the direction of vector R would be downwards.

Representation of R:
The force vector R can be represented graphically using arrows or vectors. The length of the arrow represents the magnitude of the force, and the direction of the arrow indicates the direction of the force. The vector R can also be represented mathematically using components or coordinates. In a two-dimensional Cartesian coordinate system, the vector R can be expressed as R = (Rx, Ry), where Rx and Ry are the components of the force vector along the x and y axes, respectively.

Significance of Length and Direction:
The length and direction of the force vector R are crucial in understanding and analyzing the effects of forces on objects. The length of R determines the strength of the force, while the direction of R determines the net effect of multiple forces acting on an object. By considering the length and direction of R, engineers and physicists can calculate various parameters such as the resultant force, equilibrium conditions, and motion of objects.

In conclusion, the force vector R has a length equal to the magnitude of the force and a specific direction. This understanding of the length and direction of R is essential for accurately representing and analyzing forces in physics and engineering.

Communitive law: A.B = B.A

The commutative law states that the order of multiplication does not affect the result. In other words, when multiplying two quantities, the order in which they are multiplied does not matter.

For example, let's consider two quantities A and B. According to the commutative law, A.B is equal to B.A. This means that if we multiply A by B, we will get the same result as when we multiply B by A.

To understand this concept, let's consider a simple numerical example. Let A = 2 and B = 3.

If we multiply A by B, we get A.B = 2 * 3 = 6.
Similarly, if we multiply B by A, we get B.A = 3 * 2 = 6.

As we can see, the result is the same regardless of the order of multiplication. This holds true for any values of A and B.

In the context of mechanical engineering, the commutative law is applicable when dealing with various mathematical operations, such as vector algebra and matrix multiplication. It allows us to rearrange the order of operations without changing the final result.

The commutative law is a fundamental property of multiplication and is widely used in various fields of science and mathematics. It simplifies calculations and allows for more efficient problem-solving. However, it is important to note that the commutative law does not apply to all mathematical operations. For example, subtraction and division do not follow the commutative property.

In conclusion, the statement "Communitive law: A.B = B.A" is correct. The commutative law of multiplication states that the order of multiplication does not affect the result.

The Cross Product and Moment of Force

The cross product is a mathematical operation that combines two vectors to produce a third vector that is perpendicular to both of the original vectors. In the context of mechanics, the cross product is often used to calculate the moment of a force about a given axis.

The Cross Product Table

When performing cross product calculations, a 3x3 matrix known as the cross product table is used. This matrix is made up of three rows and three columns, and each element represents a specific coordinate.

The elements of the cross product table are arranged in a specific order to ensure the correct calculation of the cross product. In the context of calculating the moment of a force about an axis, the elements in the cross product table represent the following:

- The first row represents the coordinates of the axis.
- The second row represents the coordinates of the point where the force is applied.
- The third row represents the coordinates of the force itself.

The Correct Order of Elements in the Cross Product Table

According to the question, the correct order of elements in the cross product table, from top to bottom, is:

a) Axis coordinates, point coordinates, and force coordinates

This means that the first row of the cross product table represents the coordinates of the axis, the second row represents the coordinates of the point where the force is applied, and the third row represents the coordinates of the force itself.

Why Option 'A' is the Correct Answer

The correct answer, option 'A', is derived from the correct order of elements in the cross product table. The order of the elements is crucial for correctly calculating the moment of a force about an axis. If the order is changed, the calculation will yield incorrect results.

Therefore, when calculating the moment of a force about an axis, it is essential to arrange the elements in the cross product table in the correct order, with the axis coordinates in the first row, the point coordinates in the second row, and the force coordinates in the third row.

The letter "M" is the 13th letter of the English alphabet. It is a consonant and is pronounced as "em". In mathematics, "M" is often used to represent variables or constants. In Roman numerals, "M" represents the number 1000.

If one is heading towards positive X-axis and the other is in the other direction opposite to the first one, with both having the same length and having an angle between them being obtuse, means that the direction is to be in the direction of either 1st quadrant or in the 4th quadrant.

B

The statement is false. The axis vector in the calculation of the moment along the axis of rotation is not necessarily collinear with the force vector. The moment is a vector quantity that describes the rotational effect of a force about a particular axis of rotation. It is given by the cross product of the position vector and the force vector.

The moment vector is perpendicular to the plane formed by the position vector and the force vector, and its direction is determined by the right-hand rule. The axis of rotation is defined as the line along which the moment vector lies. It is important to note that the axis of rotation is not necessarily collinear with the force vector.

Explanation:
- The moment vector is given by the equation:
- \(\text{Moment} = \text{Position Vector} \times \text{Force Vector}\)
- The position vector is a vector that points from the axis of rotation to the point where the force is applied. It defines the location of the force with respect to the axis of rotation.
- The force vector is a vector that represents the magnitude and direction of the force being applied.
- The cross product of the position vector and the force vector gives the moment vector, which describes the rotational effect of the force about the axis of rotation.
- Since the moment vector is perpendicular to the plane formed by the position vector and the force vector, the axis of rotation is also perpendicular to this plane.
- Therefore, the axis of rotation is not necessarily collinear with the force vector. It depends on the orientation of the position vector and the force vector relative to each other.
- In general, the axis of rotation can be any line that is perpendicular to the plane formed by the position vector and the force vector, and it passes through the point of rotation.
- So, the statement that the axis vector in the calculation of the moment along the axis of rotation is collinear with the force vector is false.

Θ where θ is the angle between the two vectors.
b) The direction of the cross product is perpendicular to both A and B, following the right-hand rule.