The resultant of two forces 3P and 2P is R. If the first force is doubled then the resultant is also doubled, then the angle between two forces is:

• A). 60°
• B). 120°
• C). 70°
• D). 180°

To find the angle between the two forces, we can use the law of cosines for vector addition. Let's denote the two forces as F1 = 3P and F2 = 2P. The resultant R can be expressed as:

Finding the Resultant

The formula for the resultant of two forces is given by:

R = √(F1² + F2² + 2 * F1 * F2 * cos(θ))

Substituting the values, we have:

R = √((3P)² + (2P)² + 2 * (3P) * (2P) * cos(θ))

This simplifies to:

R = √(9P² + 4P² + 12P² * cos(θ))

R = √(13P² + 12P² * cos(θ))

Doubling the Forces

When the first force is doubled, F1 becomes 6P. The new resultant R' is:

R' = √((6P)² + (2P)² + 2 * (6P) * (2P) * cos(θ))

This simplifies to:

R' = √(36P² + 4P² + 24P² * cos(θ))

R' = √(40P² + 24P² * cos(θ))

Setting Up the Equation

According to the problem, the new resultant is double the original:

R' = 2R

Substituting the expressions for R and R', we get:

√(40P² + 24P² * cos(θ)) = 2 * √(13P² + 12P² * cos(θ))

Squaring Both Sides

Squaring both sides eliminates the square roots:

40P² + 24P² * cos(θ) = 4 * (13P² + 12P² * cos(θ))

This expands to:

40P² + 24P² * cos(θ) = 52P² + 48P² * cos(θ)

Rearranging the Equation

Now, rearranging gives:

40P² - 52P² = 48P² * cos(θ) - 24P² * cos(θ)

-12P² = 24P² * cos(θ)

Dividing both sides by 12P² leads to:

-1 = 2 * cos(θ)

Thus, we find:

cos(θ) = -0.5

Determining the Angle

The angle θ that corresponds to cos(θ) = -0.5 is:

θ = 120°

Final Answer

The angle between the two forces is 120°.