Two ropes are attached to a tree, and forces of F⃗ 1=2.0iˆ+4.0jˆN and F⃗ 2=3.0iˆ+6.0jˆN are applied. The forces are coplanar (in the same plane). (a) What is the resultant (net force) of these two force vectors? (b) Find the magnitude and direction of this net force.

A) The resultant ( net force ) = 5i + 10 j

B) The magnitude and direction of this net force

Given that :

The vector forces are coplanar ( In the same plane and direction )

A) resultant force = (2i + 4j) + (3i + 6j) = ( 2 + 3 ) i  +  ( 4 + 6 ) j

                                                        = ( 5 i  + 10 j )

B ) Calculate The magnitude and direction of the net force

i) Magnitude of the force

R² = ( i² + j² )

    = ( 5² + 10² )

    = 25 + 100

∴ R = √ (25 + 100) = 5√5

ii) Determine the direction of resultant force

Tan ∅ = opposite / adjacent   ( vertical force ( j ) / horizontal force ( i ) )

         =  10 / 5 = 2

∅ = arctan ( 2 ) ≈ 63.4°

∴ direction of resultant force = 163.4° NE

Hence we can conclude that the resultant ( net force ) = 5i + 10 j  and The magnitude and direction of this net force

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We can see that  the 2 forces are being applied in the same direction

So the resultant force will be larger than the given forces

Resultant force = (2i + 4j) + (3i + 6j)

R = 5i + 10j

Magnitude of the resultant force :

R² = i² + j²

R² = 25 + 100

R = =

Direction of the resultant force:

Tan Θ = Vertical component of force / Horizontal component of force

Tan Θ = 10 / 5

Tan Θ = 2

Θ = Arctan (2)

Θ = 63.4 degrees

Direction is 63.4 degrees in the NE direction

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