Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 60.0∘60.0∘. If Rover exerts a force of 270 N and Fido exerts a force of 300 N, find the magnitude of the resultant force and the angle it makes with Rover’s rope
Answers
Answer:
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(i.) Therefore the resultant of Rover's rope and Fido's rope is 493.86 N.
(ii.) The angle between the resultant force and the Rover's rope is 28.26°.
Given:
The force exerted by Rover = 270 N
The force exerted by Fido = 300 N
The angle between both the ropes = 60°
To Find:
(i.) The magnitude of the resultant force
(ii.) The angle between the Rover's rope and the resultant.
Solution:
The given question can be answered as shown below.
Given that,
The force exerted by Rover F₁ = 270 N
The force exerted by Fido F₂ = 300 N
The angle between both the ropes θ = 60°
(i.) The resultant between any 2 forces is given by,
⇒ F = √ (F₁² + F₂² + 2F₁F₂cosθ )
⇒ F = √ ( 300² + 270² + ( 2 × 300 × 270 × cos 60° ) )
⇒ F = √( 90,000 + 72,900 + 81,000 )
⇒ F = √2,43,900 = 493.86
∴ The resultant Force = 493.86 N
(ii.) The angle between the resultant force and the Rover's rope,
⇒ Tan θ = ( F₂ Sin 60° ) / ( F₁ + F₂ Cos 60° )
⇒ Tan θ = ( 270 × √3/2 ) / ( 300 + ( 270 × 1/2) )
⇒ Tan θ = ( 135 × √3 ) / ( 300 + 135 )
⇒ Tan θ = 135√3 / 435
⇒ θ = Tan⁻¹ ( 135√3 / 435 ) ⇒ θ = 28.26°
∴ The angle between the resultant and the Rover's rope = 28.26°
(i.) Therefore the resultant of Rover's rope and Fido's rope is 493.86 N.
(ii.) The angle between the resultant force and the Rover's rope is 28.26°.
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