Question

Calculate the work done in the following situations. A constant force \(\mathbf{F}=\langle 2,4,1\rangle\) (in newtons) moves an object from (0,0,0) to (2,4,6) . (Distance is measured in meters.)

Short answer

Answer: The work done is 26 Joules.

Step-by-step solution

Find the displacement vector

To calculate the object's displacement vector, we must subtract the initial position from the final position. Using the subtraction formula for vectors, let's do it: \(\Delta \mathbf{r} = \mathbf{r}_f - \mathbf{r}_i\), where \(\mathbf{r}_f\) is the final position and \(\mathbf{r}_i\) is the initial position: \(\Delta \mathbf{r} = \langle 2,4,6 \rangle - \langle 0,0,0 \rangle = \langle 2,4,6 \rangle\)

Calculate the dot product of the force and displacement vectors

Now, we need to calculate the dot product of the force vector and the displacement vector. We do this by multiplying the corresponding vector components and then summing up the products: \(W = \mathbf{F} \cdot \Delta \mathbf{r} = \langle 2, 4, 1 \rangle \cdot \langle 2, 4, 6 \rangle\) \(W = (2 * 2) + (4 * 4) + (1 * 6)\)

Calculate Work Done

Now, we carry out the calculations to find the work done by the force, \(W = 4 + 16 + 6 = 26\) Joules The work done by the force in moving the object from (0,0,0) to (2,4,6) is 26 Joules.

Key concepts

Vector Calculus

Vector calculus is a branch of mathematics that deals with multivariable functions and three-dimensional vectors. It is instrumental in physics, especially when dealing with quantities that have both magnitude and direction, such as force and velocity.

• Vectors: Represent quantities having direction and magnitude.
• Scalar: Numerical values without direction, such as temperature or mass.

In our exercise, vector calculus helped us express both force and displacement in vector form, making it easier to calculate work done.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. This operation combines two vectors and results in a scalar.

• Formula: If \(\mathbf{A} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{B} = \langle b_1, b_2, b_3 \rangle\), then the dot product \(\mathbf{A} \cdot \mathbf{B}\) is calculated as \(a_1 \times b_1 + a_2 \times b_2 + a_3 \times b_3\).
• Significance: The dot product is useful in physics for finding how much of one vector goes in the direction of another.

In the given exercise, the dot product enabled us to calculate work, which is a scalar quantity derived from the multiplication of vectors.

Displacement Vector

A displacement vector represents the change in position of an object. It is a vector that starts at an object's initial position and ends at its final position, showcasing both distance and direction.

• Formula: The displacement vector \(\Delta \mathbf{r}\) is found by subtracting the initial position \(\mathbf{r}_i\) from the final position \(\mathbf{r}_f\), yielding \(\Delta \mathbf{r} = \mathbf{r}_f - \mathbf{r}_i\).
• Importance: It provides a direct route from start to finish, ignoring the path taken.

In our exercise, we calculated the displacement vector as \(\langle 2, 4, 6 \rangle\), indicating the movement of the object from point \(\langle 0, 0, 0 \rangle\) to \(\langle 2, 4, 6 \rangle\).

Force Vector

A force vector is a vector that shows the influence a force can have on an object's motion. It is a combination of the force's magnitude and its direction. In physics, force vectors are crucial for understanding dynamics.

• Components: In three dimensions, a force vector \(\mathbf{F}\) can be written as \(\langle F_x, F_y, F_z \rangle\), where each component acts along a respective axis.
• Role in Work: The force vector's alignment with the displacement vector determines how much work is done by the force.

In this exercise, the force vector \(\langle 2, 4, 1 \rangle\) interacted with the displacement vector to compute the work done, reflecting both its strength and the direction of the force applied.