Problem 60
Question
A force is given by the vector \(\mathbf{F}=5 \mathbf{i}+7 \mathbf{j} .\) The force moves an object along a straight line from the point \((8,11)\) to the point \((18,20)\). Find the work done if the distance is measured in meters and the force is measured in newtons.
Step-by-Step Solution
Verified Answer
The work done is 113 Joules.
1Step 1: Identify the force and position vectors
The force vector \(\mathbf{F}\) is given by \(\mathbf{F} = 5 \mathbf{i} + 7 \mathbf{j}\). The initial and final points are (8,11) and (18,20) respectively. So, the initial position can be represented as \(\mathbf{r}_i = 8 \mathbf{i} + 11 \mathbf{j}\) and the final position as \(\mathbf{r}_f = 18 \mathbf{i} + 20 \mathbf{j}\).
2Step 2: Calculate the displacement vector
The displacement is the difference between the final and initial position vectors. So, \(\mathbf{d} = \mathbf{r}_f - \mathbf{r}_i = (18 \mathbf{i} + 20 \mathbf{j}) - (8 \mathbf{i} + 11 \mathbf{j}) = 10 \mathbf{i} + 9 \mathbf{j}\).
3Step 3: Compute the dot product of force and displacement vectors
\(\mathbf{F} \cdot \mathbf{d} = (5 \mathbf{i} + 7 \mathbf{j}) \cdot (10 \mathbf{i} + 9 \mathbf{j}) = 5*10 + 7*9 = 113 \, \text{Joules}\). The units are Joules since 1 Newton meter is equal to 1 Joule.
Key Concepts
Vector OperationsDot ProductDisplacement in PhysicsForce in Physics
Vector Operations
Vectors are mathematical entities that have both magnitude and direction. They are incredibly useful in physics for representing quantities like force and displacement. To perform vector operations, we usually use vector addition and subtraction. For example, to find the resultant vector in an operation, you can add or subtract the corresponding components.
- Addition: If you have vectors \( \mathbf{A} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{B} = b_1 \mathbf{i} + b_2 \mathbf{j} \), their sum is \( \mathbf{A} + \mathbf{B} = (a_1 + b_1) \mathbf{i} + (a_2 + b_2) \mathbf{j} \).
- Subtraction: The difference is \( \mathbf{A} - \mathbf{B} = (a_1 - b_1) \mathbf{i} + (a_2 - b_2) \mathbf{j} \).
Dot Product
The dot product, also known as the scalar product, is an operation that takes two vectors and returns a single scalar. This operation is highly valuable in physics, especially when calculating work done.
The dot product is given by the formula:
\[ \mathbf{A} \cdot \mathbf{B} = a_1 \cdot b_1 + a_2 \cdot b_2 \] for two-dimensional vectors. In the provided solution, the dot product was used to compute the work done by multiplying the corresponding components of the force and displacement vectors, and summing the results.
The resulting scalar tells us how much of one vector goes in the direction of the other.
The dot product is given by the formula:
\[ \mathbf{A} \cdot \mathbf{B} = a_1 \cdot b_1 + a_2 \cdot b_2 \] for two-dimensional vectors. In the provided solution, the dot product was used to compute the work done by multiplying the corresponding components of the force and displacement vectors, and summing the results.
The resulting scalar tells us how much of one vector goes in the direction of the other.
Displacement in Physics
Displacement is a vector that points from the initial position to the final position of an object. It isn't just the distance traveled, but rather the change in position, and it includes both magnitude and direction.
To calculate displacement, you need to find the difference between the final position vector and the initial position vector. In this case, the initial and final positions are given as \( (8, 11) \) and \( (18, 20) \) respectively.
The displacement vector is calculated as follows:
To calculate displacement, you need to find the difference between the final position vector and the initial position vector. In this case, the initial and final positions are given as \( (8, 11) \) and \( (18, 20) \) respectively.
The displacement vector is calculated as follows:
- Subtract: \( \mathbf{d} = \mathbf{r}_f - \mathbf{r}_i = (18 \mathbf{i} + 20 \mathbf{j}) - (8 \mathbf{i} + 11 \mathbf{j}) \)
- Result: This simplifies to \( 10 \mathbf{i} + 9 \mathbf{j} \).
Force in Physics
In physics, force is a vector quantity that causes an object to undergo a change in speed, direction, or shape. It is typically represented by vectors because direction is an integral part of its behavior.
In the context of the exercise, the force vector \( \mathbf{F} = 5 \mathbf{i} + 7 \mathbf{j} \) was given, which means the force has a component of 5 units in the x-direction and 7 units in the y-direction.
In the context of the exercise, the force vector \( \mathbf{F} = 5 \mathbf{i} + 7 \mathbf{j} \) was given, which means the force has a component of 5 units in the x-direction and 7 units in the y-direction.
- Units: Force is measured in Newtons (N).
- Role in Work: The force vector is key to calculating work, as work is essentially the result of a force acting on an object along a displacement.
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