Equilibrium MCQ Quiz - Objective Question with Answer for Equilibrium - Download Free PDF

Explanation:

Collinear Forces

• Collinear forces act along the same line of action.

• Example: Forces on a rope in a tug of war, where all the forces are aligned along the rope.

Coplanar Concurrent Forces 

• These forces lie in the same plane and meet at a common point.

• Example: Forces on a rod resting against a wall, where normal reaction, weight, and friction act in one plane and meet at a point.

Additional InformationNon-Coplanar Concurrent Forces 

• These forces act in different planes but still intersect at a single point.

• Example: A tripod carrying a camera, where the three legs exert forces in different directions but all converge at the camera mount.

Non-Coplanar Parallel Forces → 4

• Forces that are parallel but not in the same plane.

• Example: The weight of benches in a classroom, where each bench’s weight acts vertically but at different locations and planes.

Explanation:

Varignon's Theorem:

• It states that the moment of a resultant force about any point is equal to the algebraic sum of the moments of its component forces about the same point.
• This is applicable in both 2D and 3D force systems and is fundamental in statics.
• It is valid for coplanar as well as spatial force systems and is widely used in structural and mechanical analysis.
• It supports the concept of equilibrium by ensuring that replacing a system of forces with their resultant maintains the same turning effect.

 Additional Information

Lami's Theorem:

• Applies only to a particle in equilibrium acted upon by three concurrent, coplanar, and non-parallel forces.
• It relates the magnitudes of the forces to the sine of angles between them.
•  

Cauchy's Theorem:

• Generally used in the context of stress analysis and tensor calculus, not in basic statics or moment calculation.
• This is primarily used in the theory of elasticity and stress tensors.
• It explains how the stress vector on a plane within a body can be expressed using the stress tensor.

Euler's Theorem:

• Refers to various results in mechanics and mathematics. In mechanics, it may relate to Euler's equations of motion or the rotation of rigid bodies, not directly to moments of force systems.
• In mechanics, Euler’s Theorem relates to rotation of rigid bodies and the Euler angles.
• It also refers to Euler's equations of motion in rotational dynamics and stability analysis.

Concept:

To find the resultant of collinear forces, algebraic sum is used. Forces in the same direction are added, and those in the opposite direction are subtracted.

Given:

Three forces: 250 N → right, 150 N → left (opposite direction), 350 N → right

Calculation:

Net Resultant Force = 250 + 350 - 150 = 450 N

Hence, the resultant force is: 450 N

Explanation:

Two Equal and Opposite Forces at a Different Point on a Rigid Body

• When two equal and opposite forces are applied at different points on a rigid body, the net force on the body remains zero because these forces cancel each other out. However, the purpose of introducing such forces is not to alter the net force but to transfer or relocate the original force to a different position on the body without changing its magnitude or direction. This principle is widely used in mechanics for simplifying the analysis of forces acting on rigid bodies.

Moment of a Force:

• A force applied to a rigid body causes both translational and rotational motion. The rotational effect of a force about a point or axis is termed the "moment of a force" (or torque). The moment is mathematically expressed as:

Moment (M) = Force (F) × Perpendicular Distance (d)

Where:

• F = Magnitude of the force
• d = Perpendicular distance between the line of action of the force and the axis/point of rotation

When two equal and opposite forces are introduced at different points, their line of action is such that the forces form a couple. A couple produces a pure rotational effect (moment) without causing any translational motion.

Why Transfer Forces?

In many practical engineering problems, it is convenient to transfer the point of application of a force to simplify the analysis. For instance:

• In structures, to analyze load distribution across beams and columns, the forces are often relocated for ease of calculation.
• In machines, forces are transferred to determine the resultant effect on different components.
• In robotics, force transfer helps in understanding the resultant torque on joints and links.

This transfer is achieved by introducing a pair of equal and opposite forces (forming a couple) such that the net force remains unchanged, and the original force is effectively relocated to the desired point.

Concept:

Principal of Resolution:

• It states, "The algebraic sum of the resolved parts of a number of forces in a given direction is equal to the resolved part of their resultant in the same direction."

Resolution of a Force:

When a force is resolved into two mutually perpendicular directions, without changing its effect on the body, the parts along those directions are called resolved parts. And this process is called the Resolution of a force.

Horizontal component (∑H) = Pcosθ 

Vertical component (∑V) = Psinθ 

Horizontal component (∑H) = Psinθ 

Vertical component (∑V) = Pcosθ

When a force is resolved along two mutually perpendicular directions, such as the x-axis and y-axis, the components can be determined using trigonometric functions. This method is both simple and effective because it leverages the orthogonality of the axes, ensuring that the components do not interfere with each other.

Explanation:

Free-Body Diagram: These are the diagrams used to show the relative magnitude and direction of all external forces acting upon an object in a given situation. A free-body diagram is a special example of vector diagram.

Some common rules for making a free-body diagram:

• The size of the arrow in a free-body diagram reflects the magnitude of the force.
• The direction of the arrow shows the direction that the force is acting.
• Each force arrow in the diagram is labeled to indicate the exact type of force.
• It is generally customary in a free-body diagram to represent the object by a box and to draw the force arrow from the center of the box outward in the direction that the force is acting.

Example:

Concept:

Velocity ratio in a pulley system:

• The ratio of the distance moved by the effort force applied to the object and the distance moved by the object under load is known as the Velocity ratio of the pulley system.

Velocity ratio = \(\frac{Distance~travelled~by~the~effort}{Distance~travelled~by~the~load}\)

Mechanical Advantage of pulley system:

• Mechanical Advantage = efficiency × Velocity ratio

Calculation:

Given:

Efficiency, η = 60 %

Velocity ratio = \(\frac{Distance~travelled~by~the~effort}{Distance~travelled~by~the~load}\) = \(\frac{12}{3}\) = 4

Mechanical Advantage = efficiency × Velocity ratio = 0.6 × 4 = 2.4

Additional InformationEfficiency:

• It is a measure of performance and effectiveness of a system or component.
• The main approach to define efficiency is the ratio of useful output per required input.

Mechanical Advantage:

• Mechanical Advantage is the ratio of load to effort.
• Pulleys and levers alike rely on mechanical advantage.
• The larger the advantage is the easier it will be to lift the weight.
• The mechanical advantage (MA) of a pulley system is equal to the number of ropes supporting the movable load.

Concept:

\(Moment = P × OC\)

And

\(Area\;of\;triangle = \frac{1}{2} × AB × OC\)

                            \( = \frac{1}{2} × P × OC\)

                            = \(\frac{1}{2}\)× moment

∴ Moment = Twice the area of a triangle

Concept:

Conditions for the system to be in equlibrium

ΣF_x = 0, ΣF_y = 0, ΣM = 0

Calculation:

Given:

m = 2 kg, Assume g = 10 m / s²

Lager will be the moment, smaller will be the force required to lift the rod. Hence, Applying Moment about 0 cm point we get.

w × 50 = F × 100

m × g × 50 = F × 100

2 × 10 × 50 = F × 100

F = 10 N

Concept:

Lami's Theorem: It is an equation that relates the magnitude of the three co-planner, concurrent and non-collinear forces that keeps a body in equilibrium. It states that each force is proportional to the sine of the angle between the other two forces.

Calculation:

\(\frac{{{{\rm{T}}_1}}}{{\sin 120^\circ }} = \frac{{{{\rm{T}}_2}}}{{\sin 150^\circ }} = \frac{{500}}{{\sin 90^\circ }}\)

T₁ = 500 × sin 120° and  T₂ = 500 sin 150°

T₁ = 433 N and T₂ = 250 N

Explanation:

Type of Connection	Reaction	Number of Unknowns
Weight legs link		One - The reaction is a force that acts in the direction of the link
Rollers		One - The reaction is a force that act perpendicular to the surface at point of contact.
Pin or Hinge		Two - The reaction are two force components
Guided rollar/ Fixed connected collar		Two - The reactions are a force and a moment
Fixed support		Three - The reactions are two forces and a moment
Pin connected collar		One - The reaction is a force that acts perpendicular to the surface at the point of contact

Explanation:

Given:

∠BAC = 60° 

∠BCA = 30° 

Let F_AB = compressive 

F_BC = compressive

F_AB = ?

Considering joint B,

∑F_H = 0 

F_AB cos 60° = F_BC cos 30° 

\(F_{AB} = \sqrt{3} \ F_{BC}\)

∑F_V = 0 

F_AB sin 60° + F_BC sin 30° = 10

\(\dfrac{\sqrt{3}}{2} F_{AB} + \dfrac{F_{AB}}{\sqrt{3}} \times \dfrac{1}{2} = 10\)

Multiplying both sides by \(\sqrt {3}\)

\(3 F_{AB} + F_{AB} = 20\sqrt{3}\)

\(4F_{AB} = 20\sqrt{3}\)

\(F_{AB} = 5 \sqrt{3}\)

Compressive (As direction assumed comp)

Explanation:

To determine the couple force exerted by the blade edges on the screw slot, we need to calculate the force applied by the screwdriver blade and then multiply it by the lever arm.

The formula for torque is given by:

Torque = Force x Lever Arm

In this case, the torque is 4 Nm and the width of the screwdriver blade is 5 mm. However, we need to convert the width of the blade to meters before proceeding with the calculation. 1 mm is equal to 0.001 meters.

Width of the screwdriver blade = 5 mm = 5 x 0.001 m = 0.005 m

Now we can rearrange the formula for torque to solve for force:

Force = Torque / Lever Arm

Force = 4 Nm / 0.005 m = 800 N

Therefore, the couple force exerted by the blade edges on the screw slot is 800 N.

So, the correct answer is option 2.

Concept:

Lami's theorem:

Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear forces, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding forces. According to the theorem:

\(\frac{A}{{\sin \alpha }} = \frac{B}{{\sin\beta }} = \frac{C}{{\sin\gamma }}\)

Calculation:

Given:

From the given figure we have 

\(\frac{T_1}{sin~(120)}~=~\frac{T_2}{sin~(150)}~=~\frac{T_3}{sin~(90)}\)

By solving the above equation we have,

\(\frac{T_1}{T_2}~=~\sqrt3\) and \(\frac{T_1}{T_3}~={\frac{\sqrt{3}}{2}}\)

CONCEPT:

Law of Parallelogram of forces: This law is used to determine the resultant of two coplanar forces acting at a point.

• It states that “If two forces acting at a point are represented in magnitude and direction by two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes through that common point.”

Let two forces F1 and F2, acting at the point O be represented, in magnitude and direction, by the directed line OA and OB inclined at an angle θ with each other.

Then if the parallelogram OACB be completed, the resultant force R will be represented by the diagonal OC.

\(R = \sqrt{F_1^2 + F_2^2 + 2{F_1}{F_2}cosθ } \)

CALCULATION:

Given F1 = P, F2 = √2P, θ = 135^∘ 

Then the resultant force is given by 

\(F_{total} = \sqrt{P^{2}+(\sqrt2P)^{2}+2\times P\times \sqrt 2P\times cos135^∘}\)

\(F_{total} = \sqrt{P^2+2P^{2}-2P^{2}} = P\)