Two 20.-newton forces act concurrently on
an object. What angle between these forces will produce a resultant force with the greatest magnitude?
(1) 0°
(2) 45°
(3) 90.°
(4) 180.°
20 + 20 = 40 N. @ 0o.
Answer = 0o.
To determine the angle between the forces that will produce a resultant force with the greatest magnitude, we need to consider the concept of vector addition.
When two forces act concurrently, the resultant force is obtained by adding the two forces vectorially. The magnitude of the resultant force is given by the formula:
Resultant force = √(F₁² + F₂² + 2F₁F₂cosθ)
where F₁ and F₂ are the magnitudes of the two forces, and θ is the angle between them.
To maximize the magnitude of the resultant force, we need to find the maximum value of the expression √(F₁² + F₂² + 2F₁F₂cosθ).
By observing the formula, we can see that the term containing the angle θ is the cosine term (2F₁F₂cosθ). The magnitude of the resultant force will be maximum when the cosine term is equal to 1 because cosθ is maximum when θ is 0°.
Therefore, the angle between the forces that will produce the resultant force with the greatest magnitude is 0°.
So, the correct option is (1) 0°.
To find the angle between the forces that will produce the resultant force with the greatest magnitude, we need to analyze the concept of vector summation.
When two forces act concurrently, their resultant force can be found using vector addition. The magnitude of the resultant force is given by the formula:
Resultant Force = √((Force1)^2 + (Force2)^2 + 2 × Force1 × Force2 × cosθ)
Where Force1 and Force2 are the magnitudes of the two forces, and θ is the angle between them.
To determine the angle that produces the greatest magnitude for the resultant force, we need to maximize this formula. Since the magnitudes of Force1 and Force2 are both 20N in this case, we can simplify the formula to:
Resultant Force = √(20^2 + 20^2 + 2 × 20 × 20 × cosθ)
Simplifying further:
Resultant Force = √(400 + 400 + 800 × cosθ)
Resultant Force = √(800 + 800 × cosθ)
To maximize the resultant force, we need to maximize the expression √(800 + 800 × cosθ). Since the maximum value of cosθ is 1, we can substitute it into the equation to find the maximum resultant force:
Resultant Force = √(800 + 800 × 1)
Resultant Force = √(800 + 800)
Resultant Force = √1600
Resultant Force = 40N
To find the angle, we need to solve for cosθ:
40 = √(800 + 800 × cosθ)
Squaring both sides:
1600 = 800 + 800 × cosθ
800 × cosθ = 800
cosθ = 1
Given that cosθ = 1, the angle θ that produces the greatest magnitude for the resultant force is 0°. Therefore, the correct option is (1) 0°.