Problem 79
Question
The forces \(\mathbf{F}_{1}, \mathbf{F}_{2}, \mathbf{F}_{3}, \ldots, \mathbf{F}_{n}\) acting on an object are in equilibrium if the resultant force is the zero vector: $$ \mathbf{F}_{1}+\mathbf{F}_{2}+\mathbf{F}_{3}+\cdots+\mathbf{F}_{n}=\mathbf{0} $$ In Exercises \(79-82,\) the given forces are acting on an object. a. Find the resultant force. b. What addirional force is required for the given forces to be in equilibrium? $$ \mathbf{F}_{1}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{F}_{2}=6 \mathbf{i}+2 \mathbf{j} $$
Step-by-Step Solution
Verified Answer
The resultant force is \( 9\mathbf{i}-3\mathbf{j} \) and the equilibrium force is \( -9\mathbf{i}+3\mathbf{j} \)
1Step 1: Find the resultant force
When vectors are added or subtracted, the result is a new vector. The resultant force \( \mathbf{F}_{r} \) can be found by adding \( \mathbf{F}_{1} \) and \( \mathbf{F}_{2} \): \( \mathbf{F}_{r}=\mathbf{F}_{1}+\mathbf{F}_{2}=(3 \mathbf{i}-5 \mathbf{j})+(6 \mathbf{i}+2 \mathbf{j})=(3+6) \mathbf{i}+(-5+2)\mathbf{j}=9\mathbf{i}-3\mathbf{j} \)
2Step 2: Find the equilibrium force
To find the equilibrium force, we must find the force that will bring the resultant force to zero. If the resultant force and the equilibrium force are added together, the result should be \( \mathbf{0} \). So, we subtract the resultant force from zero to find the equilibrium force \( \mathbf{F}_{e} \): \( \mathbf{F}_{e}=\mathbf{0}-\mathbf{F}_{r}=-(9\mathbf{i}-3\mathbf{j})=-9\mathbf{i}+3\mathbf{j} \)
Key Concepts
Resultant ForceEquilibrium in PhysicsVector Addition
Resultant Force
The concept of resultant force is central in understanding how various forces interact with each other when applied to an object. A resultant force is the single force which represents the vector sum of all individual forces acting on an object. It effectively combines all the different forces and their respective directions into one force which has the equivalent effect on the object as all the original forces combined.
For example, if two forces are acting upon an object, one could be pushing or pulling in an easterly direction, while the other could be in a northerly direction. The resultant force will then be neither purely east nor north but somewhere in between – precisely calculated using vector addition, which we will discuss later. The larger the magnitude of a force, the greater its influence on the resultant force's direction and size.
In our textbook example, the resultant force \( \mathbf{F}_r =\mathbf{F}_1+\mathbf{F}_2 \) is calculated by summing up the individual vectors, giving us a new vector \( 9\mathbf{i}-3\mathbf{j} \). This resultant tells us the overall effect of the combined forces on the object in terms of both magnitude and direction.
For example, if two forces are acting upon an object, one could be pushing or pulling in an easterly direction, while the other could be in a northerly direction. The resultant force will then be neither purely east nor north but somewhere in between – precisely calculated using vector addition, which we will discuss later. The larger the magnitude of a force, the greater its influence on the resultant force's direction and size.
In our textbook example, the resultant force \( \mathbf{F}_r =\mathbf{F}_1+\mathbf{F}_2 \) is calculated by summing up the individual vectors, giving us a new vector \( 9\mathbf{i}-3\mathbf{j} \). This resultant tells us the overall effect of the combined forces on the object in terms of both magnitude and direction.
Equilibrium in Physics
Moving on to equilibrium in physics, it is a condition where all the forces acting on an object are balanced, resulting in a state of no change in motion. There are two types of equilibrium: static, where the object is at rest, and dynamic, where the object is moving with a constant velocity. In both cases, the key is that there is no net force acting upon the object, meaning the resultant force is zero.
This state of balance is crucial in understanding and predicting the behavior of objects. When forces are in equilibrium, an object will not accelerate; that is, it will not start moving if at rest, or it will not change its speed or direction if in motion. In the exercise provided, Equilibrium occurs when \( \mathbf{F}_r + \mathbf{F}_e = \mathbf{0} \) which implies that the equilibrium force must be equal in magnitude and opposite in direction to the resultant force.
The exercise challenges us to find this force, essentially asking, 'What additional force would make the object's motion unaffected by the original forces?' By calculating \( \mathbf{F}_e = -\mathbf{F}_r \) we get \( -9\mathbf{i}+3\mathbf{j} \) illustrating that to achieve equilibrium, this is the force that we must apply.
This state of balance is crucial in understanding and predicting the behavior of objects. When forces are in equilibrium, an object will not accelerate; that is, it will not start moving if at rest, or it will not change its speed or direction if in motion. In the exercise provided, Equilibrium occurs when \( \mathbf{F}_r + \mathbf{F}_e = \mathbf{0} \) which implies that the equilibrium force must be equal in magnitude and opposite in direction to the resultant force.
The exercise challenges us to find this force, essentially asking, 'What additional force would make the object's motion unaffected by the original forces?' By calculating \( \mathbf{F}_e = -\mathbf{F}_r \) we get \( -9\mathbf{i}+3\mathbf{j} \) illustrating that to achieve equilibrium, this is the force that we must apply.
Vector Addition
Lastly, let's dive into the concept of vector addition. Vectors represent quantities that have both magnitude and direction, which is why they are more complex to deal with than mere numbers (scalars). When adding vectors, we must consider both aspects. Unlike scalars, where addition is straightforward, vector addition involves summing up both the horizontal (x-axis) and vertical (y-axis) components separately.
To add vectors graphically, one method is the 'head-to-tail' method, where the head (the arrowhead of a vector) is placed at the tail (the starting point) of another vector. The resultant vector is then drawn from the tail of the first vector to the head of the last. Algebraically, as in the textbook exercise, you sum up the corresponding components. For vectors in two dimensions, this would typically look like: \( (a_1\mathbf{i} + b_1\mathbf{j}) + (a_2\mathbf{i} + b_2\mathbf{j}) = (a_1 + a_2)\mathbf{i} + (b_1 + b_2)\mathbf{j} \).
In the given example, vector addition helps us to establish the resultant force by adding up \( \mathbf{F}_1 \) and \( \mathbf{F}_2 \) component-wise, leading us to the simplified expression of \( 9\mathbf{i}-3\mathbf{j} \). Understanding vector addition is crucial for breaking down complex problems into more manageable parts, crucial for students who aspire to master physics or engineering.
To add vectors graphically, one method is the 'head-to-tail' method, where the head (the arrowhead of a vector) is placed at the tail (the starting point) of another vector. The resultant vector is then drawn from the tail of the first vector to the head of the last. Algebraically, as in the textbook exercise, you sum up the corresponding components. For vectors in two dimensions, this would typically look like: \( (a_1\mathbf{i} + b_1\mathbf{j}) + (a_2\mathbf{i} + b_2\mathbf{j}) = (a_1 + a_2)\mathbf{i} + (b_1 + b_2)\mathbf{j} \).
In the given example, vector addition helps us to establish the resultant force by adding up \( \mathbf{F}_1 \) and \( \mathbf{F}_2 \) component-wise, leading us to the simplified expression of \( 9\mathbf{i}-3\mathbf{j} \). Understanding vector addition is crucial for breaking down complex problems into more manageable parts, crucial for students who aspire to master physics or engineering.
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