1. Two tugboats pull a barge across the harbor. one boat exerts of force of 7.5 x 10^4 N North, while the second boat exerts a force of 9.5 x 10^4 N at 15.0 degrees north of west. Precisely, in what direction does the barge move?
2. A traffic signal is supported by two cables, each of which makes an angle of 40.0 degrees with the vertical. If each cable can exert a maximum force of 7.50 x 10^2 N, what is the largest weight they can support?
1. It moves in the direction of the resultant force
2. Write an equation that says that the resultant vertical force is zero. With a bit of trigonometry, that will let you solve for the cable tension.
Please show your work
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1. To determine the precise direction in which the barge moves, we need to find the resultant force acting on it. This can be done by using vector addition.
First, we need to convert the force exerted by the second boat into its horizontal and vertical components. Since the force is given at an angle north of west, we can use trigonometry to find these components.
The horizontal component (F_x) can be calculated using the formula F_x = F * cos(theta), where F is the magnitude of the force and theta is the angle north of west. In this case, F = 9.5 x 10^4 N and theta = 15.0 degrees. Thus, F_x = 9.5 x 10^4 N * cos(15.0 degrees).
The vertical component (F_y) can be calculated using the formula F_y = F * sin(theta), where F is the magnitude of the force and theta is the angle north of west. In this case, F = 9.5 x 10^4 N and theta = 15.0 degrees. Thus, F_y = 9.5 x 10^4 N * sin(15.0 degrees).
Now, we can add the horizontal components and vertical components of both forces to find the resultant force. The horizontal component in the north direction is simply the sum of the horizontal components of the two forces. The vertical component in the north direction is the sum of the vertical components of the two forces.
Finally, we can use trigonometry again to find the direction of the resultant force. The angle can be calculated using the formula theta = atan(F_y / F_x), where F_x and F_y are the horizontal and vertical components of the resultant force.
2. To find the largest weight the cables can support, we need to determine the total tension in the cables. Since each cable makes an angle of 40.0 degrees with the vertical, we can use trigonometry to find the vertical and horizontal components of the tension.
The vertical component of the tension in each cable is given by T * sin(theta), where T is the magnitude of the tension and theta is the angle between the cable and the vertical. In this case, T = 7.50 x 10^2 N and theta = 40.0 degrees. The vertical component is T * sin(40.0 degrees).
The total vertical tension is twice the vertical component of tension since there are two cables supporting the weight.
Finally, we can compute the largest weight that the cables can support by dividing the total vertical tension by the acceleration due to gravity (g ≈ 9.8 m/s^2), as weight is equal to mass times gravity.
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