Can Vector Addition of Forces Result in Equilibrium?

[Masq]

Homework Statement

1.Show using a force triangle that the vector addition of the 3 forces is zero.
2. Calculate the X and Y component of forces and show that for each set of data:
ΣFx=0
ΣFy=0

Trial A: Mass(g) Direction (°)
F1 (Pulley#1): 50g 200°
F2 (Pulley#2): 95g 0°
F3 (Pulley#3): 50g 160°

Homework Equations

I think you use Newton's Second Law?

The Attempt at a Solution

1) Calculate the forces of each pulley
F1=
F2=
F3=
ΣF= F1+F2+F3=0
*How do I calculate the force of the pulley?

2)
* will edit the rest in later as I figure it out :P

 

[anathema]

Hello, great question! To show that the vector addition of the 3 forces is zero, we can use a force triangle. A force triangle is a graphical representation of the forces acting on an object, where the length and direction of each side represents the magnitude and direction of the force.

For this problem, we can start by drawing a triangle and labeling the sides with the given forces and directions. We can then use basic trigonometry to calculate the x and y components of each force. The x component can be found by multiplying the force by the cosine of the direction angle, and the y component can be found by multiplying the force by the sine of the direction angle.

Once we have calculated the x and y components for each force, we can add them together to find the total x and y components (ΣFx and ΣFy). If ΣFx and ΣFy both equal zero, this means that the vector addition of the forces is zero, as the x and y components cancel each other out.

To calculate the force of each pulley, we can use Newton's Second Law, which states that the net force on an object is equal to the mass of the object multiplied by its acceleration (F=ma). In this case, the acceleration is zero as the object is not moving, so the net force is also zero. We can set up equations for each pulley and solve for the force.

I hope this helps! Let me know if you have any further questions.

 

[baba89]

I would like to clarify that the force triangle method is a graphical method for vector addition and is not a formal proof. It is used as a visual aid to understand the concept of vector addition. In order to prove that the vector addition of the three forces is zero, we need to use mathematical equations.

Firstly, to calculate the force of each pulley, we need to use the formula F = m x a, where F is the force, m is the mass and a is the acceleration. In this case, the acceleration is due to gravity, which is 9.8 m/s^2.

F1 = (50g)(9.8 m/s^2) = 490 N
F2 = (95g)(9.8 m/s^2) = 931 N
F3 = (50g)(9.8 m/s^2) = 490 N

Now, to show that the vector addition of the three forces is zero, we need to calculate the X and Y components of each force. The X component is calculated as Fcosθ and the Y component is calculated as Fsinθ, where θ is the direction in degrees.

For F1: FX = F1cos200° = -F1cos(180°+20°) = -F1cos20° = -490cos20° = -468.6 N
FY = F1sin200° = F1sin(180°+20°) = F1sin20° = 490sin20° = 168.9 N

For F2: FX = F2cos0° = F2 = 931 N
FY = F2sin0° = 0 N

For F3: FX = F3cos160° = -F3cos(180°-20°) = -F3cos20° = -490cos20° = -468.6 N
FY = F3sin160° = F3sin(180°-20°) = F3sin20° = 490sin20° = 168.9 N

Now, we can see that the X and Y components of each force add up to zero, as shown below:
ΣFx = -468.6 N + 931 N - 468.6 N = 0
ΣFy = 168.9 N + 0 N + 168.