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. $$\mathbf{F}=\langle x, 2\rangle \text { from } A(0,0) \text { to } B(2,4)$$

Short answer

Answer: Yes, the force field is conservative and the work done to move an object between points A and B is $\frac{9}{2}$.

Step-by-step solution

Determine the work done

To find the work done, we will use the line integral formula for work done in a force field, which is given by: $$ W = \int_C \mathbf{F} \cdot d\mathbf{r}, $$ where \(W\) is the work done, \(\mathbf{F}\) is the force field, and \(C\) is the curve. In this case, the force field is \(\mathbf{F}=\langle x, 2\rangle\) and the line segment from A to B is represented as a curve, parameterized as: $$ \mathbf{r}(t) = \langle t, 2t \rangle, $$ where \(0 \leq t \leq 1\). Now, we need to differentiate the parameterization with respect to \(t\) to find \(d\mathbf{r}\): $$d\mathbf{r} = \frac{d\mathbf{r}}{dt} dt = \langle 1, 2 \rangle dt$$ Next, we need to compute the dot product \(\mathbf{F} \cdot d\mathbf{r}\): $$ \begin{aligned} \mathbf{F} \cdot d\mathbf{r} &= \langle x, 2 \rangle \cdot \langle 1, 2 \rangle dt \\ &= (x)(1) + (2)(2) dt \\ &= (x + 4) dt \end{aligned} $$ Now, we can plug this into the line integral and evaluate it: $$ \begin{aligned} W &= \int_0^1 (x+4) dt \\ &= \int_0^1 (t+4) dt \end{aligned} $$ By integrating, we have: $$ \begin{aligned} W &= \left[\frac{1}{2}t^2 + 4t \right]_0^1 \\ &= \left(\frac{1}{2}(1)^2 + 4(1)\right) - \left(\frac{1}{2}(0)^2 + 4(0)\right) \\ &= \frac{1}{2} + 4 \\ &= \frac{9}{2} \end{aligned} $$ Thus, the work done to move the object from point A to point B in this force field is \(\frac{9}{2}\).

Check if the force field is conservative

A force field is conservative if the curl of the force field is zero. The curl of a force field \(\mathbf{F}=\langle M, N \rangle\) is given by: $$ \text{Curl}(\mathbf{F}) = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} $$ In this case, \(\mathbf{F}=\langle x, 2 \rangle\), so \(M=x\) and \(N=2\). Now, we find the partial derivatives: $$ \begin{aligned} \frac{\partial N}{\partial x} &= \frac{\partial(2)}{\partial x} = 0, \\ \frac{\partial M}{\partial y} &= \frac{\partial(x)}{\partial y} = 0. \end{aligned} $$ Therefore, the curl of the force field is: $$ \text{Curl}(\mathbf{F}) = 0 - 0 = 0 $$ Since the curl of the force field is zero, the force field is conservative. In conclusion, the work required to move an object in the force field \(\mathbf{F}=\langle x, 2\rangle\) along the line segment between the given points A(0,0) and B(2,4) is \(\frac{9}{2}\), and the force field is conservative.

Key concepts

Work Done by a Force Field

When an object moves through a force field, work is done on the object by the force field. In calculus, we calculate this using the line integral of the force field along the path of the object. The general formula for the work done, W, is given by \[ W = \[\int_C \mathbf{F} \cdot d\mathbf{r} \], \]where \( \mathbf{F} \) is the force field, C is the curve or the path taken by the object, and \( d\mathbf{r} \) is a differential element of the curve C. In practical terms, you multiply the force vector by the displacement vector at every infinitesimal segment of the curve and sum up all these tiny amounts of work across the path.

Conservative Force Field

A conservative force field is one in which the work done by the force field on an object moving between two points is independent of the path taken. Mathematically, a force field is conservative if its curl is zero. The curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. In two dimensions, if curl of the force field \( \mathbf{F} \) is zero, it implies that the force field can be written as the gradient of a potential function. This is significant because it means that in a conservative force field, the work done around any closed loop is zero, leading to the conservation of mechanical energy within the system. The zero curl condition is a quick check to determine the conservativeness of a field, making it a useful tool in many physics and engineering problems.

Parameterization of a Curve

Parameterizing a curve means describing the curve using a parameter, usually denoted as \( t \), which varies over some interval. The curve is then represented by a vector function of t, \( \mathbf{r}(t) \), where each value of t corresponds to a point on the curve. This representation allows us to traverse the curve and calculate various properties, such as length, and is essential when evaluating integrals over the curve, including work done by a force field. Parameterizing provides the necessary bridge between abstract geometric paths and concrete mathematical entities needed for computation.

Curl of a Vector Field

The curl of a vector field is a measure of the field's tendency to 'rotate' around a point. In a two-dimensional context, we can think of it as the amount of 'twisting' or 'circulation' the field creates around a point. It's calculated using the partial derivatives of the field components, and a zero curl indicates no net rotation, often implying that the vector field is conservative. Understanding the curl is vital for fields like fluid dynamics and electromagnetism, as it helps in assessing whether energy can be conserved in a system and determining the behavior of force fields around different objects or particles.