Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 62.0∘. Dog A exerts a force of 270N , and dog B exerts a force of 302N .

Find the magnitude of the resultant force.

Find the angle the resultant force makes with the rope of dog A.

magnitude

To find the magnitude of the resultant force, we can use the parallelogram law of vector addition. According to this law, the magnitude of the resultant force is given by:

Resultant force = √(A^2 + B^2 + 2ABcosθ)

where A is the magnitude of the force exerted by dog A (270N), B is the magnitude of the force exerted by dog B (302N), and θ is the angle between the ropes (62.0 degrees).

Substituting the given values into the equation, we have:

Resultant force = √(270^2 + 302^2 + 2*270*302*cos(62.0))
= √(72900 + 91204 + 2*270*302*cos(62.0))
= √(72900 + 91204 + 163560*cos(62.0))
= √(72900 + 91204 + 70786.919)
= √234890.919
≈ 484.6N

Therefore, the magnitude of the resultant force is approximately 484.6N.

To find the angle the resultant force makes with the rope of dog A, we can use the sine rule. According to the sine rule,

sin(θ) / B = sin(α) / Resultant force

where θ is the angle between the resultant force and the rope of dog A (which we want to find), B is the magnitude of the force exerted by dog B (302N), and α is the angle between dog B's rope and the resultant force.

Rearranging the equation, we have:

sin(θ) = (B * sin(α)) / Resultant force

Substituting the given values, we have:

sin(θ) = (302 * sin(62.0)) / 484.6
= (302 * 0.8837) / 484.6
= 0.5493

Taking the arcsine of both sides, we get:

θ ≈ arcsin(0.5493)
≈ 33.8 degrees

Therefore, the angle the resultant force makes with the rope of dog A is approximately 33.8 degrees.

To solve this problem, we can use vector addition. The resultant force is the vector sum of the forces applied by both Dog A and Dog B.

Step 1: Resolve the forces into their x and y components.
The force applied by Dog A can be resolved into x and y components as follows:
Fx(A) = 270 N * cos(0°) = 270 N (cos(0°) = 1)
Fy(A) = 270 N * sin(0°) = 0 N (sin(0°) = 0)

The force applied by Dog B can be resolved into x and y components as follows:
Fx(B) = 302 N * cos(62°)
Fy(B) = 302 N * sin(62°)

Step 2: Add the x and y components separately.
Fx(resultant) = Fx(A) + Fx(B)
Fy(resultant) = Fy(A) + Fy(B)

Step 3: Calculate the magnitude of the resultant force.
Magnitude of the resultant force = sqrt((Fx(resultant))^2 + (Fy(resultant))^2)

Step 4: Calculate the angle the resultant force makes with the rope of Dog A.
Angle = arctan(Fy(resultant)/Fx(resultant))

Let's calculate step by step:

Step 1: Resolving the forces:
Fx(A) = 270 N * cos(0°) = 270 N
Fy(A) = 270 N * sin(0°) = 0 N

Fx(B) = 302 N * cos(62°)
Fy(B) = 302 N * sin(62°)

Step 2: Adding the components:
Fx(resultant) = 270 N + (302 N * cos(62°))
Fy(resultant) = 0 N + (302 N * sin(62°))

Step 3: Calculating the magnitude of the resultant force:
Magnitude of the resultant force = sqrt((Fx(resultant))^2 + (Fy(resultant))^2)

Step 4: Calculating the angle the resultant force makes with the rope of Dog A:
Angle = arctan(Fy(resultant)/Fx(resultant))

Now, let's substitute the values and calculate the answers.