The magnitude and direction of two forces acting on an object are 100 pounds, S79degreesE, and 60 pounds N68degreesE, respectively. Find the magnitude, to the nearest tenth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force.
find the N-S and E-W components of the two forces
add the corresponding components to find the components of the resultant
The angles are referenced to the Y-axis. so we'll assume CW rotation from +Y-axis.
F1 = 100Lbs[79o] E. of S. = 180-79 = 101o CW,
F2 = 60Lbs[68o] E. of N. = 68o CW.
Fr = 100[101o] + 60[68o] = Resultant force,
X = 100*sin101 + 60*sin68 = 153.8 Lbs.
Y = 100*Cos101 + 60*Cos68 = 3.4 Lbs.
(Fr )^2 = X^2 + Y^2 = 23,666,
Fr = 153.84 Lbs. = Resultant force.
Tan A = X/Y For CW rotation.
To find the magnitude and direction of the resultant force, you can use vector addition.
Step 1: Convert the given forces from magnitude and direction notation to component form.
The first force has a magnitude of 100 pounds and a direction of S79degreesE. To convert this into component form, we need to break it down into its horizontal and vertical components.
The horizontal component is given by:
Cosine of the angle * Magnitude = Cos(79) * 100 = -20.6 pounds
The vertical component is given by:
Sine of the angle * Magnitude = Sin(79) * 100 = -97.6 pounds
So the first force can be represented as -20.6 pounds horizontally and -97.6 pounds vertically.
The second force has a magnitude of 60 pounds and a direction of N68degreesE. To convert this into component form, we need to break it down into its horizontal and vertical components.
The horizontal component is given by:
Cosine of the angle * Magnitude = Cos(68) * 60 = 27.9 pounds
The vertical component is given by:
Sine of the angle * Magnitude = Sin(68) * 60 = 52.5 pounds
So the second force can be represented as 27.9 pounds horizontally and 52.5 pounds vertically.
Step 2: Add the horizontal and vertical components together to get the overall horizontal and vertical components of the resultant force.
The horizontal component of the resultant force is -20.6 pounds + 27.9 pounds = 7.3 pounds.
The vertical component of the resultant force is -97.6 pounds + 52.5 pounds = -45.1 pounds.
Step 3: Use the Pythagorean theorem to find the magnitude of the resultant force.
Magnitude = square root of (horizontal component squared + vertical component squared).
Magnitude = square root of (7.3 pounds squared + (-45.1 pounds) squared) = square root of (53.29 + 2034.01) = square root of 2087.3 = 45.7 pounds (rounded to the nearest tenth of a pound).
Step 4: Use trigonometry to find the direction angle of the resultant force.
The direction angle can be calculated using the arctan function.
Direction angle = arctan(vertical component / horizontal component)
Direction angle = arctan(-45.1 pounds / 7.3 pounds) = arctan(-6.17) = -80.6 degrees (rounded to the nearest tenth of a degree).
Therefore, the magnitude of the resultant force is approximately 45.7 pounds, and the direction angle is approximately -80.6 degrees.