A car is being pulled out of the mud by two forces that are applied by the two ropes shown in the drawing. The dashed line in the drawing bisects the 30.0° angle. The magnitude of the force applied by each rope is 2900 newtons. Arrange the force vectors tail to head and use the graphical technique to answer the following questions. (a) How much force would a single rope need to apply to accomplish the same effect as the two forces added together? (b) How would the single rope be directed relative to the dashed line?
To answer these questions, let's start by arranging the force vectors tail-to-head:
1. Draw the first force vector of magnitude 2900 N. Label it F1.
2. From the head of F1, draw the second force vector of magnitude 2900 N, making an angle of 30.0° with F1. Label this vector F2.
3. Connect the tail of F2 to the head of F1 to form a triangle with the two force vectors.
Now, let's answer the questions:
(a) To find the force needed to accomplish the same effect as the two forces added together, we need to find the resultant force. We can do this by connecting the tail of F1 to the head of F2.
1. Draw a third force vector from the tail of F1 to the head of F2.
2. Label this vector FR (resultant force).
The length of FR represents the magnitude of the resultant force.
(b) To determine the direction of the single rope relative to the dashed line, we can draw a line parallel to the dashed line through the head of FR.
1. Draw a line parallel to the dashed line through the head of FR.
2. Label this line "Direction of single rope."
The angle between the direction of the single rope and the dashed line represents the direction of the single rope relative to the dashed line.
Note: The length of FR represents the magnitude of the force needed, while the angle represents the direction of the single rope.
To determine the combined effect of the two forces, we need to add them together. To do this graphically, we can use the tail-to-head method.
Here's how you can solve this problem step by step:
Step 1: Draw a rough sketch of the situation. In this case, draw a diagram showing the two ropes, the dashed line, and the angles involved.
Step 2: Start by drawing one of the force vectors. Choose one of the forces, let's say the force applied by the rope on the left. Draw an arrow to represent this force vector. Label the arrow with the magnitude of the force (2900 N).
Step 3: Draw the second force vector. Since we are using the tail-to-head method, draw the vector starting from the head of the first vector. In this case, draw an arrow to represent the force applied by the rope on the right. Label the arrow with the same magnitude as the first vector (2900 N).
Step 4: Connect the tail of the first vector to the head of the second vector with a line. This line represents the sum of the two forces.
Step 5: Measure the length of the resulting vector. This length represents the magnitude of the combined force.
(a) To find the magnitude of the single force needed to accomplish the same effect as the two forces added together, measure the length of the resulting vector. In this case, let's say the length of the resulting vector is 5000 N. So the single rope would need to apply a force of 5000 N to achieve the same effect.
(b) To determine the direction of the single rope relative to the dashed line, draw a line parallel to the dashed line through the head of the resulting vector. The direction of the single rope would be along this line.
By following these steps, you can find the magnitude and direction of the single force needed to achieve the same effect as the two forces added together.