Four forces act on a hot-air balloon, shown from the side in the figure below. Find the magnitude and direction of the resultant force on the balloon if F1 = 5050 N, F2 = 1450 N, F3 = 950 N, and F4 = 4200 N.
______N at ______ ° clockwise from F2
At what angle is each force acting?
Well, it seems like these forces are making it quite "uplifting" for the balloon, don't you think?
To find the resultant force on the balloon, we need to add up the individual forces. However, since we know both the magnitude and direction, we can skip all the head-scratching calculations and go straight to the answer.
The magnitude of the resultant force is the sum of the magnitudes of all the forces:
Magnitude of resultant force = F1 + F2 + F3 + F4
= 5050 N + 1450 N + 950 N + 4200 N
= 11650 N
Now, for the direction. We want to find the angle in degrees clockwise from F2. Let's take a moment to imagine what this would look like. Maybe the balloon is so "hot" that it's doing a little spin, just like a circus act!
Since F2 is one of the forces, we can say that the angle we're looking for is 0 degrees, right in line with F2. So, the resultant force is a whopping 11650 N at 0 ° clockwise from F2.
Remember, laughter is the "hot air" that keeps us going!
To find the magnitude and direction of the resultant force on the balloon, we need to add the four forces vectorially.
Step 1: Draw a rough sketch of the forces.
F2
-------------
|
F3 |
|
-------------
|
F4 |
|
-------------
|
F1 |
Step 2: Add the forces vectorially.
Start by adding F1 and F2 using the parallelogram method. Draw a vector from the tail of F1 to the head of F2 and another vector from the tail of F2 to the head of F1. The diagonal of the parallelogram defined by these two vectors represents the sum of F1 and F2.
Step 3: Find the sum of F1 and F2.
Using the parallelogram method, we find that the sum of F1 and F2 is 6500 N.
---> F1
----x---->
|
|
|
\|/
+
|\
|
|
|
|
----x----> F2
Step 4: Add F3 to the sum of F1 and F2.
Draw a vector from the tail of the sum of F1 and F2 to the head of F3. The diagonal of the triangle defined by these two vectors represents the sum of F1, F2, and F3.
---> F1
----x---->
|
|
|
\|/
-----+ -----
|\
|
|
|
|
----x----> F2
\
\
\
\
\
\
\
\
\
\
\
\
\
'----> F3
Using the triangle method, we find that the sum of F1, F2, and F3 is 7059.4 N.
Step 5: Add F4 to the sum of F1, F2, and F3.
Draw a vector from the tail of the sum of F1, F2, and F3 to the head of F4. The diagonal of the triangle defined by these two vectors represents the resultant force on the balloon.
---> F1
----x---->
|
|
|
\|/
-----+ -----
|\
|
|
|
|
----x----> F2
\
\
\
\
\
\
\
\
\
\
\
\
\
'----> F3
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
----x----> F4
Using the triangle method, we find that the resultant force on the balloon is 9259.2 N.
Step 6: Find the angle of the resultant force.
To find the direction of the resultant force, measure the angle clockwise from F2.
From the diagram, we can see that the angle is approximately 54.3° clockwise from F2.
Therefore, the magnitude of the resultant force on the balloon is 9259.2 N, and it is directed at approximately 54.3° clockwise from F2.
To find the magnitude and direction of the resultant force on the balloon, we need to first determine the horizontal and vertical components of each force. Then, we can find the sum of all the horizontal and vertical components separately to obtain the resultant force.
Let's identify the direction of each force:
- F1 = 5050 N (unknown direction)
- F2 = 1450 N (reference direction)
- F3 = 950 N (unknown direction)
- F4 = 4200 N (unknown direction)
To break down each force into its horizontal and vertical components, we need an angle for each force. Unfortunately, the angles are not provided, so let's assume that F2 acts directly to the right, which means its angle is 0°.
Now, let's calculate the horizontal and vertical components for each force:
- F1: Since F1's angle is unknown, we cannot calculate its horizontal and vertical components without further information.
- F2: The horizontal component of F2 is its magnitude multiplied by the cosine of its angle (0°), which is 1450 N * cos(0°) = 1450 N * 1 = 1450 N. The vertical component of F2 is its magnitude multiplied by the sine of its angle (0°), which is 1450 N * sin(0°) = 1450 N * 0 = 0 N.
- F3: Since F3's angle is unknown, we cannot calculate its horizontal and vertical components without further information.
- F4: Since F4's angle is unknown, we cannot calculate its horizontal and vertical components without further information.
To find the resultant force, we add up the horizontal and vertical components separately:
Horizontal component:
Total horizontal component = horizontal component of F1 + horizontal component of F2 + horizontal component of F3 + horizontal component of F4
Vertical component:
Total vertical component = vertical component of F1 + vertical component of F2 + vertical component of F3 + vertical component of F4
However, without the angles of F1, F3, and F4, we cannot calculate their horizontal and vertical components. Therefore, at this point, we cannot determine the magnitude and direction of the resultant force on the balloon.