Two forces are pushing an ice shanty along the Ice. One has a magnitude of 330 lb in a direction due east. The other force has a magnitude of 110 lb in a direction 54 degrees east of north. What are the magnitude and direction of the resulant force?
method 1: state them as vectors in component form
first : 330 due east
= (330cos 0, 330 sin 0) = (330, 0)
second( 110cos36°, 110sin36°) = (88.99 , 64.66)
add them to get (418.99, 64.66)
magnitude = √(418.99^2 + 64.66) = 423.95
direction: tanØ = 64.66/418.99
Ø = 8.77°
or a bearing of N 8.77° E
method 2:
construction the parallogram with the diagonal representing the magnitude
I get an obtuse angled triangle with sides 110 and 330 and the contained angle of 144° ...... (90+54)
by the cosine law:
r^2 = 110^2 + 330^2 - 2(110)(330)cos144
...
r = 423.95 , same as above
Use the Sine Law to find the direction of the resultant.
To find the magnitude and direction of the resultant force, we need to use vector addition. Here's how you can do it step by step:
1. Label the given forces:
- Force A: 330 lb in the east direction
- Force B: 110 lb at an angle of 54 degrees east of north
2. Convert Force B into its horizontal and vertical components:
- Horizontal component (Bx): 110 lb * cos(54)
- Vertical component (By): 110 lb * sin(54)
3. Resolve Force A into horizontal and vertical components:
- Horizontal component (Ax): 330 lb * cos(0)
- Vertical component (Ay): 330 lb * sin(0)
4. Add the horizontal and vertical components of the forces:
- Horizontal component of the resultant force (Rx) = Ax + Bx
- Vertical component of the resultant force (Ry) = Ay + By
5. Calculate the magnitude and direction of the resultant force:
- Magnitude of the resultant force (R) = sqrt(Rx^2 + Ry^2)
- Direction of the resultant force (θ) = tan^(-1)(Ry/Rx)
Now let's calculate the resultant force:
- Bx = 110 lb * cos(54) = 59.21 lb (rounded to two decimal places)
- By = 110 lb * sin(54) = 88.17 lb (rounded to two decimal places)
- Ax = 330 lb * cos(0) = 330 lb (no rounding needed)
- Ay = 330 lb * sin(0) = 0 lb (no rounding needed)
- Rx = Ax + Bx = 330 lb + 59.21 lb = 389.21 lb (rounded to two decimal places)
- Ry = Ay + By = 0 lb + 88.17 lb = 88.17 lb (rounded to two decimal places)
To calculate the magnitude of the resultant force:
- R = sqrt(Rx^2 + Ry^2) = sqrt((389.21 lb)^2 + (88.17 lb)^2) = 403.24 lb (rounded to two decimal places)
To calculate the direction of the resultant force:
- θ = tan^(-1)(Ry/Rx) = tan^(-1)(88.17 lb / 389.21 lb) = 13.93 degrees (rounded to two decimal places)
Therefore, the magnitude of the resultant force is approximately 403.24 lb, and the direction is approximately 13.93 degrees.