two dogs pull horizontally on ropes attached to a post; the angle between the rope is 70deg. dog A exerts a force of 268N and dog B exerts a force of 310N.
find the magnitude of the resultant force. and the angle the resultant force makes with dog A`s rope.
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To solve this problem, we can use the principles of vector addition. The forces exerted by the two dogs can be represented as vectors. The magnitude of the resultant force can be found by finding the sum of these vectors, and the angle it makes with Dog A's rope can be determined using trigonometry.
Step 1: Convert the forces into vector components
Since the angles are given with respect to horizontal axes, we need to split the forces into their horizontal (x) and vertical (y) components. We can use trigonometry to do this.
For Dog A:
Force_Ax = Force_A * cos(angle_A)
Force_Ay = Force_A * sin(angle_A)
For Dog B:
Force_Bx = Force_B * cos(angle_B)
Force_By = Force_B * sin(angle_B)
Given:
Force_A = 268 N
Force_B = 310 N
angle_A = 70°
angle_B = 180° - 70° (since the angles add up to 180°)
Step 2: Calculate the vector components
Using the given values, we can calculate the x and y components for both forces:
For Dog A:
Force_Ax = 268 N * cos(70°)
Force_Ay = 268 N * sin(70°)
For Dog B:
Force_Bx = 310 N * cos(110°) (180° - 70°)
Force_By = 310 N * sin(110°)
Step 3: Find the sum of the vector components
To find the resultant force, we need to add the x and y components separately:
Resultant_Fx = Force_Ax + Force_Bx
Resultant_Fy = Force_Ay + Force_By
Step 4: Determine the magnitude of the resultant force
Using the x and y components, we can calculate the magnitude of the resultant force using the Pythagorean theorem:
Resultant_F = sqrt(Resultant_Fx^2 + Resultant_Fy^2)
Step 5: Find the angle between the resultant force and Dog A's rope
To find the angle, we can use trigonometry once again:
angle_resultant = tan^(-1)(Resultant_Fy / Resultant_Fx)
Now, let's calculate the values:
Force_Ax = 268 N * cos(70°) = -112.48 N (negative because it's in the opposite direction to the positive x-axis)
Force_Ay = 268 N * sin(70°) = 248.78 N
Force_Bx = 310 N * cos(110°) = -211.02 N
Force_By = 310 N * sin(110°) = -181.21 N
Resultant_Fx = -112.48 N + (-211.02 N) = -323.50 N
Resultant_Fy = 248.78 N + (-181.21 N) = 67.57 N
Resultant_F = sqrt((-323.50 N)^2 + (67.57 N)^2) = 337.43 N
angle_resultant = tan^(-1)(67.57 N / -323.50 N) = -11.86°
Therefore, the magnitude of the resultant force is approximately 337.43 N, and the angle the resultant force makes with Dog A's rope is approximately -11.86°.