Two and Three Force Systems MCQ Quiz - Objective Question with Answer for Two and Three Force Systems - Download Free PDF

Explanation:

Parallelogram Law of Forces

It states that if two forces acting at a point are represented in magnitude and direction by the adjacent sides of a parallelogram, then the diagonal of the parallelogram represents their resultant force.

Additional InformationKey points of Parallelogram Law of Forces

• Two Forces: The law applies to two forces acting on a body.

• Parallelogram Formation: These forces form adjacent sides of a parallelogram.

• Resultant Force: The diagonal of the parallelogram represents the magnitude and direction of the resultant force.

• Magnitude and Direction: The magnitude of the resultant force is found by using vector addition (geometrically represented by the diagonal), and its direction is the direction of the diagonal of the parallelogram.

Explanation:

Lami's theorem:

It states that if three forces acting at a point are in equilibrium, each force is proportional to the sine of the angle between the other two forces.

Consider three forces FA, FB, FC acting on a particle or rigid body making angles α, β and γ with each other.

Then from Lami's theorem,

\(\frac{{{F_A}}}{{sin\alpha }} = \frac{{{F_B}}}{{sin\beta }} = \frac{{{F_C}}}{{sin\gamma }}\)

Explanation:

Resultant between two forces P and Q, acting at an angle θ is defined as:

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 parallelogram which passes through that common point.”

Let two forces P and Q, 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.

\({\rm{R}} = \sqrt {{\rm{P}}^2 + {\rm{Q}}^2 + 2{{\rm{P}}}{{\rm{Q}}}\cos {\rm{\theta }}}\)

Explanation:

Equations of equilibrium for Non-concurrent force System: A non-concurrent force system will be in equilibrium if the resultant of all forces and moments is zero.

• ΣFx = 0, ΣFy = 0 and ΣM = 0

Equations of equilibrium for concurrent force System: For the concurrent forces, the lines of action of all forces met at a point and hence the moment of those forces about that point will be zero or ΣM = 0 automatically.

• ΣFx = 0 and ΣFy = 0

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 

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

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

Explanation:

When two force making an angle θ, the resultant (R) of the two force is given by:

\(R=\sqrt{F_1^2+F_2^2+2F_1F_2\cosθ}\;\)

Since the two forces are equal;

\(R=\sqrt{F_1^2+F_2^2+2F_1F_2\cosθ}\;\)

\(R=\sqrt{F^2+F^2+2F^2\cosθ}\;\)

\(R=\sqrt{2F^2+2F^2\cosθ}\;\)

\(R=\sqrt{2F^2(1\;+\;\cosθ)}\;\)

We know that;

1 + cos θ = 2 cos²(θ/2)

\(R=\sqrt{2F^2[2\cos^2(θ/2)]}\;\)

R = 2F cos(θ/2)

Explanation:

Principle of resolution of forces:

• 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.

Method for the resolution for the resultant force:

• Resolve all the forces horizontally and find the algebraic sum of the horizontal components.
• Resolve all the forces vertically and find the algebraic sum of the vertical components.

The resultant of the above both can be given below as,

\(R = \sqrt {{{\left( {\sum H} \right)}^2} + {{\left( {\sum V} \right)}^2}} \)

The resultant force will make an angle with the horizontal can be given as,

\(tan\theta = \frac{{\sum V}}{{\sum H}}\)

Principle of Transmissibility:

• According to it if we transmit a force in its line of action without changing its magnitude and direction then there will not be a change in the effect of a force.

Principle of Independence of force:

• It is defined as vertical motion does not affect by the movement of horizontal motion.

Concept:

Resultant between two forces P and Q, acting at an angle θ is defined as:

\(R = \sqrt {{P^2} + {Q^2} + 2\times P\times Q\times cosθ } \)

Calculation:

Given that, Resultant Force = \(60\sqrt{3}\) , the angle between them θ = 60°, also the forces are equal i.e. P = Q

 \(R = \sqrt {{P^2} + {Q^2} + 2\times P\times Q\times cosθ } \)

⇒ \(60\sqrt{3} = \sqrt {{P^2} + {P^2} + 2\times P\times P\times \cos 60} = P\sqrt 3 \)

⇒ \(P = \frac{{60\sqrt 3}}{{\sqrt 3 }} = 60\)

Concept:

Resultant of two forces F₁ & F₂ is given by;

\(R = \sqrt {F_1^2 + F_2^2 + 2{F_1}{F_2}\;cos\;θ } \)

R = Resultant Force, θ = angle beween two forces

Calculation:

Given:

F₁ = F₂, R = F₁

now, \(R = \sqrt {F_1^2 + F_2^2 + 2{F_1}{F_2}\;cos\;θ } \)

\(F_1 = \sqrt {F_1^2 + F_1^2 + 2{F_1}{F_1}\;cos\;θ } \)

F₁² = 2F₁² + 2F₁² cos θ 

2 × cos θ = -1

\(\cos θ = \; - \frac{1}{2}\)

cos θ = cos 120°

θ = 120° 

Concept:

To find the tension in a member of a truss, we can use either Lami’s Theorem or the Method of Joints. At joint P, three forces act in equilibrium: vertical load F, force in member PQ (45°), and force in member PR (30°). Since PR and QR are collinear, the force in PR is the same as the tension in QR.

Calculation:

Using Lami’s Theorem at joint P:

The angles between the forces are:

• Between PQ and PR: 45° + 30° = 75°
• Between F and PR: 60°
• Between F and PQ: 45°

Apply Lami's Theorem:

\( \frac{T_{PQ}}{\sin(60^\circ)} = \frac{T_{PR}}{\sin(45^\circ)} = \frac{F}{\sin(75^\circ)} \)

We are interested in:

\( T_{QR} = T_{PR} = \frac{F \cdot \sin(45^\circ)}{\sin(75^\circ)} \)

Now, plug in the values:

• \( \sin(45^\circ) = 0.7071 \)
• \( \sin(75^\circ) = 0.9659 \)

\( T_{QR} = \frac{F \cdot 0.7071}{0.9659} \approx 0.732 F \)

Alternate Method (Method of Joints):

Let T₁ be force in PQ (at 45°), and T₂ be force in PR (at 30°):

Vertical equilibrium:

\( T_1 \sin(45^\circ) + T_2 \sin(30^\circ) = F \)

Horizontal equilibrium:

\( T_2 \cos(30^\circ) = T_1 \cos(45^\circ) \)

\( T_1 = \frac{0.866}{0.7071} T_2 \approx 1.2247 T_2 \)

Substitute into vertical equation:

\( 1.2247 T_2 \cdot 0.7071 + T_2 \cdot 0.5 = F \)

\( T_2 (0.866 + 0.5) = F \Rightarrow T_2 = \frac{F}{1.366} \approx 0.732 F \)

Explanation:

Equations of equilibrium for Non-concurrent force System: A non-concurrent force system will be in equilibrium if the resultant of all forces and moments is zero.

• ΣFx = 0, ΣFy = 0 and ΣM = 0

Equations of equilibrium for concurrent force System: For the concurrent forces, the lines of action of all forces met at a point and hence the moment of those forces about that point will be zero or ΣM = 0 automatically.

• ΣFx = 0 and ΣFy = 0

Explanation:

When two force making an angle θ, the resultant (R) of the two force is given by:

\(R=\sqrt{F_1^2+F_2^2+2F_1F_2\cosθ}\;\)

Since the two forces are equal;

\(R=\sqrt{F_1^2+F_2^2+2F_1F_2\cosθ}\;\)

\(R=\sqrt{F^2+F^2+2F^2\cosθ}\;\)

\(R=\sqrt{2F^2+2F^2\cosθ}\;\)

\(R=\sqrt{2F^2(1\;+\;\cosθ)}\;\)

We know that;

1 + cos θ = 2 cos2(θ/2)

\(R=\sqrt{2F^2[2\cos^2(θ/2)]}\;\)

R = 2F cos(θ/2)

Here, F = P, so

R = 2P cos(θ/2)

Explanation:

Principle of resolution of forces:

• 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.

Method for the resolution for the resultant force:

• Resolve all the forces horizontally and find the algebraic sum of the horizontal components.
• Resolve all the forces vertically and find the algebraic sum of the vertical components.

The resultant of the above both can be given below as,

\(R = \sqrt {{{\left( {\sum H} \right)}^2} + {{\left( {\sum V} \right)}^2}} \)

The resultant force will make an angle with the horizontal can be given as,

Important Points

Principle of Transmissibility:

• According to it if we transmit a force in its line of action without changing its magnitude and direction then there will not be a change in the effect of a force.

Principle of Independence of force:

• It is defined as vertical motion does not affect by the movement of horizontal motion.

Concept:

Let the length of the rod be 'L' and 'x' be the distance between the resultant force and smaller force, therefore the distance between the larger force and resultant will be (L - x)

According to the principle of moments:

"The algebraic sum of the moments of all the forces about any point is equal to the moment of their resultant force about the same point.”

Calculation:

Given:

F₁ = 10 N, F₂ = 40 N, L = 120 mm.

When we calculate the moment about the point of the resultant force, then the moment produced by two forces will be equal and opposite.

F1 × x = F2 × (L - x)

10 × x = 40 × (120 - x)

10x = 4800 - 40x

50x = 4800

x = 96 mm.

∴ the distance between the smaller force and the resultant is 96 mm.