Problem 82 - University Physics with Modern Physics

Question

(a) Is the force \(\overrightarrow{\boldsymbol{F}}=\boldsymbol{C} y^{2} \hat{\boldsymbol{j}},\) where \(\boldsymbol{C}\) is a negative constant with units of \(\mathrm{N} / \mathrm{m}^{2},\) conservative or nonconservative? Justify your answer.

Step-by-Step Solution

Answer

The force \( \overrightarrow{\mathbf{F}} = C y^2 \hat{\mathbf{j}} \) is conservative as its curl is zero.

Step 1: Understand the Characteristics of Conservative Forces

A force is said to be conservative if the work done by the force on an object moving from one point to another depends only on the initial and final positions, and not on the path taken. Mathematically, this condition holds if the force can be expressed as the negative gradient of a scalar potential energy function.

Step 2: Check the Condition for Conservative Forces

For a force field \( \overrightarrow{\mathbf{F}} = f(y) \hat{\mathbf{j}} \), a necessary and sufficient condition for it to be conservative is that the curl of the force field is zero, i.e., \( abla \times \overrightarrow{\mathbf{F}} = \mathbf{0} \). For a force given as \( \overrightarrow{\mathbf{F}} = C y^2 \hat{\mathbf{j}} \), express the force in terms of its components for evaluation: \( F_x = 0 \), \( F_y = C y^2 \), and \( F_z = 0 \).

Step 3: Calculate the Curl of the Force

Use the determinant form to calculate the curl:\[ abla \times \overrightarrow{\mathbf{F}} = \begin{vmatrix}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \F_x & F_y & F_z\end{vmatrix} \]Substituting the component values into the determinant gives:\[ abla \times \overrightarrow{\mathbf{F}} = \hat{\mathbf{i}} \left( \frac{\partial}{\partial y}(0) - \frac{\partial}{\partial z}(C y^2) \right) + \hat{\mathbf{j}} \left( \frac{\partial}{\partial z}(0) - \frac{\partial}{\partial x}(0) \right) + \hat{\mathbf{k}} \left( \frac{\partial}{\partial x}(C y^2) - \frac{\partial}{\partial y}(0) \right) \]This simplifies to \(-\hat{\mathbf{i}} (0) + \hat{\mathbf{j}} (0) + \hat{\mathbf{k}} (0) = \mathbf{0}\).

Step 4: Conclusion

Since the curl \( abla \times \overrightarrow{\mathbf{F}} = \mathbf{0} \), the force \( \overrightarrow{\mathbf{F}} = C y^2 \hat{\mathbf{j}} \) is indeed conservative. A conservative force can be expressed as the gradient of some potential energy function, and this property is satisfied here.

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that focuses on vectors and their derivatives and integrals. It deals with vector fields, which are functions that assign a vector to every point in space. In the context of this exercise, we deal with the vector field described by the force \(\overrightarrow{\boldsymbol{F}}=\boldsymbol{C} y^{2} \hat{\boldsymbol{j}}\).
This force is expressed in terms of unit vectors, specifically the \(\hat{\boldsymbol{j}}\) component, indicating it is directed along the y-axis. An important aspect of working with vector fields in vector calculus is determining whether a force is conservative. To do this, we use tools like the curl operation. The curl of a vector field helps us understand the field's rotation. If a vector field has zero curl, it is termed irrotational, which is an essential criterion for a conservative field.

• Vector fields can be decomposed into components along the coordinate axes.
• Curl is a vector operator that describes the infinitesimal rotation of a field.
• A zero curl indicates that a vector field may have a potential energy function associated with it.

Vector calculus provides the mathematical framework to analyze forces, helping us answer questions about conservativeness through calculations like determining the curl of the force field.

Potential Energy

Potential energy refers to the energy stored within a system due to the position or configuration of objects. In physics, when a force is conservative, it can be associated with a potential energy function. This means that the work done by or against a conservative force over any closed path is zero.
In our exercise, we need to determine if the given force \(\overrightarrow{\boldsymbol{F}}=\boldsymbol{C} y^{2} \hat{\boldsymbol{j}}\) is conservative. If it is, there exists a potential energy function \(U(y)\) such that \(\overrightarrow{\boldsymbol{F}} = -abla U\). The gradient of a function, described by \(abla\), points in the direction of the greatest rate of increase of that function and its magnitude is the rate of increase per unit distance.

• A potential energy function exists if the force is conservative.
• The work done by a conservative force depends only on the initial and final positions, not the path.
• It is expressed using the negative gradient of the potential energy function: \(\overrightarrow{\boldsymbol{F}} = -abla U\).

Understanding potential energy allows us to analyze systems where energy is conserved, giving insight into how forces influence the movement of objects within those systems.

Work-Energy Theorem

The Work-Energy Theorem provides a direct relationship between work done and the change in kinetic energy of a system. According to the theorem, the work done by all forces acting on an object equals the change in its kinetic energy.
For conservative forces, this work is also related to changes in potential energy. In our scenario, we determined that the force \(\overrightarrow{\boldsymbol{F}} = C y^2 \hat{\boldsymbol{j}}\) is conservative. Thus, the work done by this force can be converted into potential energy changes when the object changes position.

• Work done by conservative forces results in potential energy change.
• The theorem asserts that total work equals the change in kinetic energy: \(W = \Delta KE\).
• In conservative fields, total mechanical energy (potential plus kinetic) is conserved.

Understanding the Work-Energy Theorem helps us connect forces and energy transformations, illustrating how mechanical energy conservation is a powerful tool in analyzing physical systems.