HELP! ok here it goes...
Compute the resultant and the equilibrant of the following system of coplanar concurrent forces: 100lbs, 30degrees; 141.41lb, 45degrees; 100lb, 240degrees.
To compute the resultant and equilibrant of a system of coplanar concurrent forces, we need to break each force vector into its x-component and y-component. Let's break down the problem step by step:
Step 1: Convert the given forces into their x-component and y-component form.
- Force 1: 100 lbs at 30 degrees
- x-component: 100 lbs * cos(30 degrees)
- y-component: 100 lbs * sin(30 degrees)
-Force 2: 141.41 lbs at 45 degrees
- x-component: 141.41 lbs * cos(45 degrees)
- y-component: 141.41 lbs * sin(45 degrees)
-Force 3: 100 lbs at 240 degrees
- x-component: 100 lbs * cos(240 degrees)
- y-component: 100 lbs * sin(240 degrees)
Step 2: Calculate the sum of all x-components and y-components of the forces.
- Sum of x-components: Add all the x-components obtained from Step 1.
- Sum of y-components: Add all the y-components obtained from Step 1.
Step 3: Find the magnitude and direction of the resultant vector.
- Magnitude: Use the Pythagorean theorem to calculate the magnitude: sqrt((Sum of x-components)^2 + (Sum of y-components)^2)
- Direction: Use inverse tangent (arctan) with the formula: arctan((Sum of y-components) / (Sum of x-components))
Step 4: Calculate the equilibrant.
- The equilibrant is a force that will balance out the resultant, so it has the same magnitude as the resultant but acts in the opposite direction.
Now, let's calculate the forces:
Force 1:
- x-component: 100 lbs * cos(30 degrees) = 100 lbs * 0.866 = 86.6 lbs
- y-component: 100 lbs * sin(30 degrees) = 100 lbs * 0.5 = 50 lbs
Force 2:
- x-component: 141.41 lbs * cos(45 degrees) = 141.41 lbs * 0.707 = 100 lbs (approximately, rounded to nearest whole number)
- y-component: 141.41 lbs * sin(45 degrees) = 141.41 lbs * 0.707 = 100 lbs (approximately, rounded to nearest whole number)
Force 3:
- x-component: 100 lbs * cos(240 degrees) = 100 lbs * (-0.5) = -50 lbs
- y-component: 100 lbs * sin(240 degrees) = 100 lbs * (-0.866) = -86.6 lbs
Sum of x-components: 86.6 lbs + 100 lbs + (-50 lbs) = 136.6 lbs (approximately, rounded to one decimal place)
Sum of y-components: 50 lbs + 100 lbs + (-86.6 lbs) = 63.4 lbs (approximately, rounded to one decimal place)
Magnitude of resultant: sqrt((136.6 lbs)^2 + (63.4 lbs)^2) = sqrt(18716.56 + 4025.96) = sqrt(22742.52) = 150.9 lbs (approximately, rounded to one decimal place)
Direction of resultant: arctan(63.4 lbs / 136.6 lbs) = arctan(0.4641) = 25.5 degrees (approximately, rounded to one decimal place)
Equilibrant: The equilibrant force has the same magnitude as the resultant force but acts in the opposite direction. Therefore, the magnitude is also 150.9 lbs, but the direction is the opposite of the resultant direction, which is 180 degrees + 25.5 degrees = 205.5 degrees (approximately, rounded to one decimal place).
Therefore, the resultant force is approximately 150.9 lbs at an angle of 25.5 degrees, and the equilibrant force is approximately 150.9 lbs at an angle of 205.5 degrees.