Problem 25 - Thomas Calculus

Question

Kinetic energy If a variable force of magnitude \(F(x)\) moves an object of mass \(m\) along the \(x\) -axis from \(x_{1}\) to \(x_{2},\) the object's velocity \(v\) can be written as \(d x / d t\) (where \(t\) represents time). Use Newton's second law of motion \(F=m(d v / d t)\) and the Chain Rule \begin{equation}\frac{d v}{d t}=\frac{d v}{d x} \frac{d x}{d t}=v \frac{d v}{d x}\end{equation} to show that the net work done by the force in moving the object from \(x_{1}\) to \(x_{2}\) is \begin{equation}W=\int_{x_{1}}^{x_{2}} F(x) d x=\frac{1}{2} m v_{2}^{2}-\frac{1}{2} m v_{1}^{2},\end{equation} where \(v_{1}\) and \(v_{2}\) are the object's velocities at \(x_{1}\) and \(x_{2} .\) In physics, the expression \((1 / 2) m v^{2}\) is called the kinetic energy of an object of mass \(m\) moving with velocity \(v .\) Therefore, the work done by the force equals the change in the object's kinetic energy, and we can find the work by calculating this change.

Step-by-Step Solution

Answer

The net work done is equal to the change in kinetic energy: \( W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2 \).

Step 1: Understand the Problem

We need to show that the net work done by a force moving an object is equal to the change in its kinetic energy. We have expressions for net work, force, velocity, and kinetic energy.

Step 2: Use Newton's Second Law

Start with Newton's second law, which states that the force \( F \) is equal to the mass \( m \) times the acceleration, \( a \). Here, \( F = m \frac{dv}{dt} \).

Step 3: Apply the Chain Rule

Use the chain rule to express the acceleration in terms of velocity and position: \( \frac{dv}{dt} = v \frac{dv}{dx} \). Therefore, \( F = m \left( v \frac{dv}{dx} \right) \).

Step 4: Write the Expression for Work Done

The work done by the force \( F(x) \) as the object moves from \( x_1 \) to \( x_2 \) is given by the integral \( W = \int_{x_1}^{x_2} F(x) \, dx \).

Step 5: Substitute Force in Terms of Velocity

Substitute \( F = m v \frac{dv}{dx} \) into the work expression: \[ W = \int_{x_1}^{x_2} m v \frac{dv}{dx} \, dx \].

Step 6: Change Variables in the Integral

Use the substitution \( dx = \frac{1}{v} \, dv \) to transform the integral: \[ W = m \int_{v_1}^{v_2} v \, dv \].

Step 7: Evaluate the Integral

Calculate the integral: \( W = m \left[ \frac{v^2}{2} \right]_{v_1}^{v_2} \).

Step 8: Simplify to Find the Change in Kinetic Energy

Plug in the limits and simplify: \[ W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2 \]. This shows that the work done is the change in kinetic energy.

Key Concepts

Newton's Second Law

Newton's Second Law of Motion is a cornerstone of classical physics. It is best known in its formulaic form: \( F = ma \), where \( F \) represents force, \( m \) is mass, and \( a \) is acceleration. This law tells us that a force acting on an object will accelerate the object in the direction of the force. The law explains how the velocity of an object changes when a force is applied, making it crucial for understanding concepts like kinetic energy and the work-energy principle. The force can change depending on the situation, so it's vital to understand each variable's role.

• Force causes acceleration, a change in velocity over time.
• The mass of the object resists changes these changes, also known as inertia.
• Time and position (displacement) are essential in calculating acceleration.

In our context with a variable force, expressing force with respect to velocity using Newton's second law involves considering how the force changes as the object moves.

Variable Force

In the scenario where a force varies with position, known as a variable force, it becomes important to express this force accurately. A variable force \( F(x) \) means that as an object moves along a path, say along the \( x \)-axis, the magnitude of the force does not remain constant.

• The force depends on the position \( x \) of the object.
• This dependence allows us to express the work done as an integral over the path.
• It requires calculus to calculate the total work done by the force.

To determine work done by such a force, we integrate the force over the path of motion. The challenge is in linking this variable force to changes in kinetic energy, which is where the detailed application of calculus principles becomes necessary.

Chain Rule

The chain rule is a fundamental concept in calculus that allows us to differentiate complex expressions. It is particularly useful when dealing with variable forces in physics because it helps relate multiple changing quantities.To relate velocity, position, and time, the chain rule is used. Here's how:

• The acceleration \( \frac{dv}{dt} \) can be decomposed into \( v \frac{dv}{dx} \), where \( v \) is velocity and \( \frac{dv}{dx} \) is the derivative of velocity with respect to position.
• This reveals how velocity is changing as the object moves from one point to another, offering deeper insight into the force's impact on motion.
• Understanding this relationship helps in transforming expressions for force and ultimately the work done.

By utilizing the chain rule, we are able to reframe how forces interact with objects in motion, crucial for evaluating work done with variable forces.

Work-Energy Principle

The work-energy principle is an essential part of understanding how energy changes in a system. It states that the work done by the forces on an object results in a change in its kinetic energy.The formula \(W = \Delta KE = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2\) clearly expresses this principle.

• Work \( W \) is defined as the force acting on an object times the distance it moves in the direction of the force. When integrated, it accumulates all small changes over a path.
• Kinetic energy is energy due to motion, proportional to the mass and square of the velocity.
• The work-energy principle provides a bridge between forces acting on an object and the resulting motion, measured as changes in kinetic energy.

Understanding and applying the work-energy principle allows us to see how energy conservation plays out in dynamic systems, where work translates into energy changes.