A barge is hauled along a straight-line section of canal by two horses harnessed to tow ropes and walking along the tow paths on either side of the canal. Each horse pulls with a force of 955 N at an angle of 15° with the centerline of the canal. Find the sum of these two forces on the barge.
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To find the sum of the two forces on the barge, we need to resolve each force into its horizontal and vertical components.
Given:
Force of each horse = 955 N
Angle with the centerline of the canal = 15°
Step 1: Resolve the forces into horizontal and vertical components
The horizontal component can be found using the equation: F_horizontal = F * cos(angle)
The vertical component can be found using the equation: F_vertical = F * sin(angle)
For each horse:
Horizontal component of the force = 955 * cos(15°)
Vertical component of the force = 955 * sin(15°)
Step 2: Add the horizontal and vertical components separately
Add the horizontal components and vertical components of the two forces separately.
Horizontal component of the sum = (955 * cos(15°)) + (955 * cos(15°))
Vertical component of the sum = (955 * sin(15°)) + (955 * sin(15°))
Step 3: Calculate the resultant force
The resultant force can be calculated using the Pythagorean theorem.
Resultant force = sqrt((horizontal component of the sum)^2 + (vertical component of the sum)^2)
Substituting the values calculated in Step 2:
Resultant force = sqrt(((955 * cos(15°)) + (955 * cos(15°)))^2 + ((955 * sin(15°)) + (955 * sin(15°)))^2)
Simplifying this expression will give us the sum of the two forces on the barge.
To find the sum of the two forces on the barge, we need to break down each force into its horizontal and vertical components. We'll use trigonometry to do this.
First, let's find the horizontal component of each force. We can use the formula:
Horizontal component = Force * cos(angle)
For the first horse:
Horizontal component of Force 1 = 955 N * cos(15°)
For the second horse, since the angle is measured from the centerline, we need to subtract it from 180° to get the angle relative to the horizontal direction:
Horizontal component of Force 2 = 955 N * cos(180° - 15°)
Next, let's find the vertical component of each force. We can use the formula:
Vertical component = Force * sin(angle)
For the first horse:
Vertical component of Force 1 = 955 N * sin(15°)
For the second horse:
Vertical component of Force 2 = 955 N * sin(180° - 15°)
Now we can add up the horizontal components and the vertical components separately to find the total force on the barge.
Horizontal force on the barge = Horizontal component of Force 1 + Horizontal component of Force 2
Vertical force on the barge = Vertical component of Force 1 + Vertical component of Force 2
Finally, we can combine these two forces using vector addition to find the sum of the forces on the barge. The sum of the forces is given by:
Sum of forces = √((Horizontal force on the barge)^2 + (Vertical force on the barge)^2)
Calculate the values of the horizontal and vertical components for each force using the given force magnitude and angle. Then substitute these values into the formulas above to find the sum of the forces on the barge.