Question

Find the work required to move an object in the following force fields along a line segment between the given points. Check to see whether the force is conservative. $$\mathbf{F}=\langle x, y, z\rangle \text { from } A(1,2,1) \text { to } B(2,4,6)$$

Short answer

Question: Find the work required to move an object in the force field \(\mathbf{F}=\langle x, y, z\rangle\) along a line segment between points A(1,2,1) and B(2,4,6). Also, check if the force is conservative. Answer: The work required to move the object along the path is 25 units. The force is conservative since its curl is zero.

Step-by-step solution

Parameterize the path

Create a parameterization of the path by finding a function \(\mathbf{r}(t)\) that goes from point A to point B as \(t\) goes from 0 to 1: $$\mathbf{r}(t) = A + t(B - A)$$ Insert the coordinates of points A and B: $$\mathbf{r}(t) = (1, 2, 1) + t((2, 4, 6) - (1, 2, 1)) = (1, 2, 1) + t(1, 2, 5)$$ Then, $$\mathbf{r}(t) = (1 + t, 2 + 2t, 1 + 5t)$$

Compute the derivatives

Compute the derivative of the position vector \(\mathbf{r}(t)\) with respect to the parameter \(t\): $$\frac{d\mathbf{r}}{dt} = \left(\frac{d}{dt}(1 + t), \frac{d}{dt}(2 + 2t), \frac{d}{dt}(1 + 5t)\right) = (1, 2, 5)$$

Compute the work along the path

Now we will find the work required to move the object along the path by finding the integral $$W = \int_C \mathbf{F} \cdot d\mathbf{r} = \int_0^1 \mathbf{F}(x(t), y(t), z(t)) \cdot \frac{d\mathbf{r}}{dt} \, dt$$ Where \(\mathbf{F}(x(t), y(t), z(t))\) is the force vector at a point along the line segment parameterized by t. $$\mathbf{F}(\mathbf{r}(t))= \langle x(t),y(t),z(t) \rangle = \langle 1 + t, 2 + 2t, 1 + 5t \rangle$$ Calculate the dot product of this force vector with the derivative of the position vector: $$\mathbf{F}(\mathbf{r}(t)) \cdot \frac{d\mathbf{r}}{dt} = (1 + t) \cdot 1 + (2 + 2t) \cdot 2 + (1 + 5t) \cdot 5 = 1 + t + 4 + 4t + 5 + 25t = 10 + 30t$$ Now, find the integral of this expression over the interval \([0, 1]\): $$W = \int_0^1 (10 + 30t) \, dt = \left[10t + 15t^2\right]_0^1 = 10 + 15 = 25$$ Therefore, the work required to move the object along the path is 25 units.

Check if the force is conservative

Finally, we need to check if the force is conservative. We can do this by calculating the curl of the vector field \(F\): $$\operatorname{Curl}(\mathbf{F}) = (\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}) \times \langle x, y, z\rangle = \langle 0, 0, 0\rangle$$ Since the curl of the force field \(\mathbf{F}\) is zero, the force is conservative.

Key concepts

Conservative Force Fields

In physics, a force field is conservative if the work done by the force on an object moving from one point to another is independent of the path taken. This means that the work done only depends on the initial and final positions. To determine whether a force field is conservative, we can calculate its curl.The curl of a vector field is a measure of the field's rotation at a point, given mathematically by the formula: \[ abla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \]If this calculation results in a zero vector, the field is conservative.Moreover, conservative force fields are associated with potential energy, and they don't cause energy loss due to a closed loop path. This is why the concept is crucial in many physical contexts, such as gravity and electrostatics.

Work Done by a Force

Work is a measure of energy transfer when an object moves under the influence of a force. In vector calculus, the work done by a force on an object is defined by the line integral of the force along a path:\[ W = \int_C \mathbf{F} \cdot d\mathbf{r} \] In our example, the force \( \mathbf{F} = \langle x, y, z\rangle \) is applied to move an object along a line segment. To find the work, we first find the dot product of the force vector with the derivative of the path's parameterization vector.- Calculate the force at each point along the path.- Find the dot product between this force and the differential path vector.- Integrate this product over the path's parameterization interval, here from \(t = 0\) to \(t = 1\). This integral gives the total work done in moving the object. The dot product helps to determine how much of the force actually goes into moving the object along the path, focusing purely on the component of the force parallel to the motion direction.

Parameterizing a Path

Parameterization is a technique used to describe a path by a continuous set of coordinates, typically using a parameter like \(t\). This parameter runs from an initial value to a final value as the object moves from the start to the end point of the path.In our exercise, the path between points \(A(1,2,1)\) and \(B(2,4,6)\) is parameterized by: \[ \mathbf{r}(t) = (1 + t, 2 + 2t, 1 + 5t) \] Here, \(t\) varies from 0 to 1.Steps to parameterize:

• Identify the start and end points of the path.
• Calculate the differences between the corresponding coordinates of these points.
• The parameterization time \(t\) scales this difference, and when added to the starting coordinates, moves the object smoothly along the path.

The derivative of this path function gives a velocity vector, crucial for calculating work done. The method provides a mathematical way to "trace" the line from A to B, enabling calculation like line integrals.