Three horizontal forces are pulling on a
ring, at rest. F1 is 12.0 N at a 0° angle, and
F2 is 18.0 N at a 90° direction. What is the
x-component of F3?
The Correct Answer and Explanation is :
Problem Understanding:
We are dealing with three horizontal forces acting on a ring. Two of them, (F_1) and (F_2), are already known, and the third force, (F_3), is the unknown force. We are asked to determine the x-component of (F_3).
Given:
- (F_1 = 12.0 \, \text{N}) at a (0^\circ) angle (along the positive x-axis).
- (F_2 = 18.0 \, \text{N}) at a (90^\circ) angle (along the positive y-axis).
Approach:
The ring is at rest, which implies that the net force acting on it must be zero. According to Newton’s first law of motion, if the object is not accelerating, the sum of the forces in both the x-direction and the y-direction must be zero. This means that the sum of the x-components of the forces and the sum of the y-components of the forces must each equal zero.
- Force components in the x-direction:
- (F_1) has an x-component of 12.0 N (since it’s along the positive x-axis).
- (F_2) has no x-component because it acts in the y-direction.
- Let (F_3) have components (F_{3x}) in the x-direction and (F_{3y}) in the y-direction. For the ring to remain at rest, the sum of the x-components must be zero:
[
F_1 + F_{3x} = 0
]
Substituting (F_1 = 12.0 \, \text{N}), we get:
[
12.0 + F_{3x} = 0
]
Solving for (F_{3x}):
[
F_{3x} = -12.0 \, \text{N}
]
Conclusion:
The x-component of (F_3) is -12.0 N. This negative value indicates that (F_3) is acting in the negative x-direction, which is required to balance out the 12.0 N force in the positive x-direction from (F_1).
Explanation:
The net force on the ring must be zero for the ring to remain at rest, meaning that the forces in both the x and y directions must cancel each other out. Since (F_1) is directed along the positive x-axis, and (F_2) along the positive y-axis, the third force, (F_3), must counteract these forces. To balance the forces in the x-direction, (F_3) must have an equal and opposite component in the x-direction. Thus, the x-component of (F_3) must be (-12.0 \, \text{N}).