Problem 31
Question
Suppose \(\mathbf{F}\) is a conservative force field with potential function \(\phi\). In physics the function \(p=-\phi\) is called potential energy. Since \(\mathbf{F}=-\nabla p\), Newton's second law becomes $$ m \mathbf{r}^{\prime \prime}=-\nabla p \quad \text { or } \quad m \frac{d \mathbf{v}}{d t}+\nabla p=\mathbf{0} $$ By integrating \(m \frac{d v}{d t} \cdot \frac{d \mathbf{r}}{d t}+\nabla p \cdot \frac{d \mathbf{r}}{d t}=0\) with respect to \(t\), derive the law of conservation of mechanical energy: \(\frac{1}{2} m v^{2}+p=\) constant.
Step-by-Step Solution
Verified Answer
Integrating the given equation shows that total mechanical energy \(\frac{1}{2} m v^2 + p\) is conserved.
1Step 1: Start with Given Equation
The given equation from Newton's second law is \(m \frac{d \mathbf{v}}{d t} + abla p = \mathbf{0}\). We need to integrate \(m \frac{d v}{d t} \cdot \frac{d \mathbf{r}}{d t} + abla p \cdot \frac{d \mathbf{r}}{d t} = 0\) with respect to time \(t\).
2Step 2: Simplify the Expression
Recognize that \(\mathbf{v} = \frac{d \mathbf{r}}{d t}\), the velocity. Thus, the expression to integrate becomes \(m \frac{d \mathbf{v}}{d t} \cdot \mathbf{v} + abla p \cdot \mathbf{v} = 0\).
3Step 3: Integrate the Kinetic Term
The term \(m \frac{d \mathbf{v}}{d t} \cdot \mathbf{v}\) can be rewritten as \(\frac{d}{dt} \left( \frac{1}{2} m v^2 \right)\), since \(v^2\) is the dot product of velocity with itself. Thus, integrate to get \(\frac{1}{2} m v^2\).
4Step 4: Integrate the Potential Energy Term
For the potential energy term, recall that the gradient \(abla p\) is the derivative of the potential function. The integral \(\int (abla p \cdot \mathbf{v}) \, dt\) simplifies to \(-p\) because it is the anti-derivative. Negative comes from the relationship of force and potential energy, \(\mathbf{F} = -abla p\).
5Step 5: Combine Results to Show Conservation
Adding the results from steps 3 and 4, we have \(\frac{1}{2} m v^2 - p = \text{constant}\). Rearranging gives \(\frac{1}{2} m v^2 + p = \text{constant}\), which shows the conservation of mechanical energy.
Key Concepts
Conservative Force FieldPotential EnergyNewton's Second LawIntegration in Physics
Conservative Force Field
A conservative force field is one where the work done by the force on an object moving between two points depends only on the positions of the points and not on the path taken. This means that the total work done in a closed path is zero.
In other words, when an object moves in a closed loop within a conservative force field, it will return to the same mechanical state it initially had. Hence, energy is conserved.
Common examples of conservative forces include gravitational force and electrostatic force. These forces are related to potential energy, setting the groundwork for understanding how these energies interact and change with motion.
To identify a conservative force field, check if the curl of the vector field is zero. Mathematically, if the force field \( \mathbf{F} \) is conservative, then \( abla \times \mathbf{F} = \mathbf{0} \).
In other words, when an object moves in a closed loop within a conservative force field, it will return to the same mechanical state it initially had. Hence, energy is conserved.
Common examples of conservative forces include gravitational force and electrostatic force. These forces are related to potential energy, setting the groundwork for understanding how these energies interact and change with motion.
To identify a conservative force field, check if the curl of the vector field is zero. Mathematically, if the force field \( \mathbf{F} \) is conservative, then \( abla \times \mathbf{F} = \mathbf{0} \).
Potential Energy
Potential energy is related to the position of an object within a force field. It represents energy that is stored and has the potential to do work in the future. Commonly, potential energy is derived from forces like gravity and elasticity.
In terms of mathematics, if \( \phi \) is the potential function, the potential energy \( p \) is given by \( p = -\phi \). This is because the potential energy is essentially the opposite of the work done by the conservative force.
When a system transitions from a higher potential energy to a lower potential energy, it usually converts this into kinetic energy, showcasing the transfer of energy between different forms while remaining within the realm of conservative forces.
In terms of mathematics, if \( \phi \) is the potential function, the potential energy \( p \) is given by \( p = -\phi \). This is because the potential energy is essentially the opposite of the work done by the conservative force.
When a system transitions from a higher potential energy to a lower potential energy, it usually converts this into kinetic energy, showcasing the transfer of energy between different forms while remaining within the realm of conservative forces.
Newton's Second Law
Newton's Second Law provides the foundation for connecting forces and motion. It states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration: \( \mathbf{F} = m \mathbf{a} \).
In a conservative force field, this law is expressed as \( m \mathbf{r}^{\prime \prime} = -abla p \). Here, \( abla p \) represents the gradient of the potential energy, acting as the conservative force that influences the object's acceleration.
Through integrating Newton's Second Law, we derive the conservation of mechanical energy, which balances kinetic and potential energy in a system under the influence of conservative forces.
In a conservative force field, this law is expressed as \( m \mathbf{r}^{\prime \prime} = -abla p \). Here, \( abla p \) represents the gradient of the potential energy, acting as the conservative force that influences the object's acceleration.
Through integrating Newton's Second Law, we derive the conservation of mechanical energy, which balances kinetic and potential energy in a system under the influence of conservative forces.
Integration in Physics
Integration is a mathematical tool used to determine the total effect of small, cumulative changes. In physics, especially in the context of motion and forces, it helps untangle complex relationships among quantities such as velocity, acceleration, and force.
For the scenario involving the conservation of mechanical energy, we integrate the terms related to kinetic and potential energy over time. This process brings together how these energies shift but sum up to a constant, indicating that the total mechanical energy remains unchanged.
The operation \( m \frac{d \mathbf{v}}{d t} \cdot \mathbf{v} \) simplifies to the kinetic energy change, \( \frac{1}{2} m v^2 \), and \( abla p \cdot \mathbf{v} \) connects back to potential energy, reminding us of the unmistakable tie between force fields and energy conservation.
Integration thus is indispensable in showing how dynamic physical quantities interrelate and obey fundamental laws like the conservation of energy.
For the scenario involving the conservation of mechanical energy, we integrate the terms related to kinetic and potential energy over time. This process brings together how these energies shift but sum up to a constant, indicating that the total mechanical energy remains unchanged.
The operation \( m \frac{d \mathbf{v}}{d t} \cdot \mathbf{v} \) simplifies to the kinetic energy change, \( \frac{1}{2} m v^2 \), and \( abla p \cdot \mathbf{v} \) connects back to potential energy, reminding us of the unmistakable tie between force fields and energy conservation.
Integration thus is indispensable in showing how dynamic physical quantities interrelate and obey fundamental laws like the conservation of energy.
Other exercises in this chapter
Problem 31
Find the directional derivative(s) of \(f(x, y)=x+y^{2}\) at \((3,4)\) in the direction of a tangent vector to the graph of \(2 x^{2}+y^{2}=9\) at \((2,1)\)
View solution Problem 31
Evaluate the given iterated integral by changing to polar coordinates. $$ \int_{-5}^{5} \int_{0}^{\sqrt{25-x^{2}}}(4 x+3 y) d y d x $$
View solution Problem 31
Verify the given identity. Assume continuity of all partial derivatives. \(\operatorname{div}(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot \operatorname{curl}
View solution Problem 31
Find the directional derivatives(s) of \(f(x, y)=x+y^{2}\) at \((3,4)\) in the direction of a tangent vector to the graph of \(2 x^{2}+y^{2}=9\) at \((2,1)\).
View solution