A 12 kilogram tire is to be pulled by three ropes. One force, F1, has a magnitude of 50N. Orient the other two forces so that the magnitude of the resulting acceleration of the tire is least. And find that magnitude if: F2=30N, F3= 20N.
To determine the orientation of the other two forces (F2 and F3) that will result in the least magnitude of acceleration for the tire, we can use the concepts of vector addition and the principle of least action. Let's break down the steps:
Step 1: Understand the problem
We have a tire with a mass of 12 kg and three pulling forces: F1 = 50 N, F2 = 30 N, and F3 = 20 N. Our goal is to find the arrangement (orientation) of F2 and F3 that will result in the least magnitude of acceleration for the tire.
Step 2: Analyze the forces
The total force acting on the tire is the vector sum of F1, F2, and F3. We can calculate the net force using the formula:
Net force (Fnet) = F1 + F2 + F3
Step 3: Understand acceleration
According to Newton's second law of motion, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The formula for acceleration is:
Acceleration (a) = Fnet / mass
Step 4: Find the orientation with the least magnitude of acceleration
To find the orientation that results in the least acceleration, we need to minimize the magnitude of the net force (Fnet). Since Fnet is a vector sum, its magnitude can be minimized by minimizing the angle between F2 and F3.
Step 5: Calculate the net force and acceleration
Using the given values, we can substitute the force values into the formulas to find the net force and acceleration:
Fnet = F1 + F2 + F3
Fnet = 50 N + 30 N + 20 N
Fnet = 100 N
Acceleration (a) = Fnet / mass
Acceleration (a) = 100 N / 12 kg
Acceleration (a) ≈ 8.33 m/s^2
So, regardless of the orientation of F2 and F3, the magnitude of the resulting acceleration for the tire is approximately 8.33 m/s^2.