Two tugboats are towing as barge. Each exerts a force of 5.0 tons and the angle between two ropes is 30 degrees. Find the resultant force exerted the barge.
Fr = (5[30o] + 5[0o] = (5*Cos30+5) + (5*sin30)I = 9.33 + 2.5i = 9.66[15o].
To find the resultant force exerted on the barge, we need to use the concept of vector addition.
Step 1: Convert the forces to a common unit
Since the given forces are in tons, let's convert them to a more common unit of force, such as Newtons. 1 ton is equal to 9,800 Newtons. So, each tugboat exerts a force of 5.0 tons = 5.0 x 9,800 = 49,000 Newtons.
Step 2: Resolve the forces into their x and y components
Since the angle between the two ropes is given as 30 degrees, we can resolve each force into its x and y components using trigonometry. The x-component is given by Fx = F * cos(theta), and the y-component is given by Fy = F * sin(theta).
For each tugboat, the x-component would be Fx = 49,000 * cos(30°) and the y-component would be Fy = 49,000 * sin(30°).
Step 3: Add the x-components and y-components separately
Now, we can add the x-components of the forces together and separately add the y-components together. Let's call the resultant force in the x-direction Rx and the resultant force in the y-direction Ry.
So, Rx = (49,000 * cos(30°)) + (49,000 * cos(30°)), and Ry = (49,000 * sin(30°)) + (49,000 * sin(30°)).
Step 4: Calculate the magnitude and direction of the resultant force
Using the x and y components, we can find the magnitude of the resultant force using the Pythagorean theorem:
Resultant force (R) = sqrt(Rx^2 + Ry^2)
And the direction of the resultant force can be found using the inverse tangent:
Direction (θ) = atan(Ry / Rx)
Now you can plug in the values and calculate the resultant force exerted on the barge.