Additional: Work-Energy Theorem and Power

Work-Energy Theorem ($ W_{net} = \Delta KE $)

The Work-Energy Theorem is a fundamental principle in physics that directly links the concepts of work and kinetic energy. It provides a powerful tool for analyzing the motion of objects, stating how the total work done on an object affects its speed.

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Statement of the Work-Energy Theorem

The theorem states that the net work done by all forces acting on an object is equal to the change in its kinetic energy.

Mathematically, this is expressed as:

$ W_{net} = \Delta KE $

where:

• $W_{net}$ is the total work done by all forces acting on the object. This is the scalar sum of the work done by each individual force, or the work done by the net force ($\vec{F}_{net} = \sum \vec{F}_i$).
• $\Delta KE$ is the change in kinetic energy of the object, calculated as $KE_{final} - KE_{initial}$.
• $KE_{final} = \frac{1}{2} m v_{final}^2$ is the kinetic energy at the end of the displacement.
• $KE_{initial} = \frac{1}{2} m v_{initial}^2$ is the kinetic energy at the beginning of the displacement.

So, $ W_{net} = \frac{1}{2} m v_{final}^2 - \frac{1}{2} m v_{initial}^2 $

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Implications of the Work-Energy Theorem

• If $W_{net}$ is positive, $\Delta KE$ is positive, meaning the kinetic energy and speed of the object increase. The net force has done positive work, accelerating the object.
• If $W_{net}$ is negative, $\Delta KE$ is negative, meaning the kinetic energy and speed of the object decrease. The net force has done negative work, decelerating the object.
• If $W_{net}$ is zero, $\Delta KE$ is zero, meaning the kinetic energy and speed of the object remain constant. The net force does no work (either zero net force, or net force is perpendicular to displacement).

The Work-Energy Theorem applies regardless of whether the forces are constant or variable, or whether the path is straight or curved. However, the derivation is simpler for a constant force along a straight line.

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Derivation of the Work-Energy Theorem (for Constant Net Force in 1D)

Consider an object of mass $m$ moving along a straight line (say, the x-axis) under the action of a constant net force $F_{net}$. Let the object's initial velocity be $v_i$ and final velocity be $v_f$ after undergoing a displacement $\Delta x$.

According to Newton's Second Law, the constant net force produces a constant acceleration $a$:

$ F_{net} = ma $

The work done by the constant net force over the displacement $\Delta x$ (assuming force and displacement are in the same direction, $\theta = 0^\circ$) is:

$ W_{net} = F_{net} \Delta x = (ma) \Delta x $

From the kinematic equation relating initial velocity ($v_i$), final velocity ($v_f$), constant acceleration ($a$), and displacement ($\Delta x$):

$ v_f^2 = v_i^2 + 2a \Delta x $

We can rearrange this equation to solve for $a \Delta x$:

$ v_f^2 - v_i^2 = 2a \Delta x $

$ a \Delta x = \frac{v_f^2 - v_i^2}{2} $

Now, substitute this expression for $a \Delta x$ back into the work done equation:

$ W_{net} = m (a \Delta x) = m \left(\frac{v_f^2 - v_i^2}{2}\right) $

$ W_{net} = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 $

Recognizing the terms as the final and initial kinetic energies, we get:

$ W_{net} = KE_{final} - KE_{initial} = \Delta KE $

This derivation confirms the Work-Energy Theorem for the case of a constant net force acting in one dimension. The theorem is more general and holds true for variable forces and in three dimensions, though the derivation requires calculus.

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Conservative and Non-Conservative Forces

Forces in physics can be broadly classified into two categories based on how the work they do depends on the path taken by the object:

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Conservative Forces

A force is called conservative if the work done by the force in moving an object from one point to another depends only on the initial and final positions of the object, and is independent of the path taken between these points.

Equivalently, a force is conservative if the work done by the force in moving an object along any closed path (starting and ending at the same point) is zero.

For a conservative force, it is possible to define a corresponding potential energy. The work done by a conservative force is equal to the negative of the change in the associated potential energy:

$ W_c = - \Delta PE = -(PE_{final} - PE_{initial}) = PE_{initial} - PE_{final} $

This means that as a conservative force does positive work (e.g., gravity pulling an object down), the potential energy decreases. As work is done against a conservative force (e.g., lifting an object up against gravity), potential energy increases.

Examples of Conservative Forces:

• Gravitational force
• Elastic spring force (for an ideal spring obeying Hooke's Law)
• Electrostatic force between stationary charges

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Non-Conservative Forces

A force is called non-conservative if the work done by the force in moving an object from one point to another depends on the path taken between these points.

Equivalently, a force is non-conservative if the work done by the force in moving an object along a closed path is not zero.

Non-conservative forces typically dissipate mechanical energy from a system, often converting it into heat or sound. A potential energy function cannot be defined for a non-conservative force in the same way it can for a conservative force.

Examples of Non-Conservative Forces:

• Frictional force (kinetic and static)
• Air resistance or drag force
• Tension in a string (when the length changes or there is friction)
• Applied force exerted by a person or motor (these are often doing work against other forces)

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Distinction and Conservation of Energy

The distinction between conservative and non-conservative forces is crucial for the concept of conservation of mechanical energy. The principle of conservation of mechanical energy ($KE + PE = \text{constant}$) holds true only when the net work done by all non-conservative forces is zero.

If non-conservative forces do work, the total mechanical energy of the system changes. The work done by non-conservative forces ($W_{nc}$) is related to the change in both kinetic and potential energy:

$ W_{nc} = \Delta (KE + PE) = \Delta E_{mechanical} $

where $E_{mechanical} = KE + PE$. This equation essentially says that non-conservative forces are responsible for the change in the total mechanical energy of a system.

However, the total energy of an isolated system (including all forms, like thermal energy generated by friction) is always conserved. Non-conservative forces simply transform mechanical energy into other forms (like heat), rather than destroying it.

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Power as Force times Velocity ($ P = \vec{F} \cdot \vec{v} $)

We previously defined power as the rate at which work is done ($P = dW/dt$) or energy is transferred. A useful alternative expression for instantaneous power relates it directly to the force acting on an object and its instantaneous velocity.

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Derivation of $P = \vec{F} \cdot \vec{v}$

Consider a force $\vec{F}$ acting on an object. Let the object undergo a small displacement $d\vec{r}$ in a small time interval $dt$.

The infinitesimal work done by the force during this small displacement is:

$ dW = \vec{F} \cdot d\vec{r} $

The instantaneous power $P$ is the rate of doing work, which is the work done $dW$ divided by the time interval $dt$:

$ P = \frac{dW}{dt} $

Substitute the expression for $dW$ into the equation for power:

$ P = \frac{\vec{F} \cdot d\vec{r}}{dt} $

Using the properties of the dot product and the definition of instantaneous velocity $\vec{v} = \frac{d\vec{r}}{dt}$, we can write:

$ P = \vec{F} \cdot \left(\frac{d\vec{r}}{dt}\right) $

$ P = \vec{F} \cdot \vec{v} $

This formula, $P = \vec{F} \cdot \vec{v}$, gives the instantaneous power delivered by the force $\vec{F}$ to the object that has instantaneous velocity $\vec{v}$.

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Physical Interpretation

Since the dot product can be written as $P = F v \cos\theta$, where $\theta$ is the angle between $\vec{F}$ and $\vec{v}$, this formula shows:

• Only the component of the force that is parallel to the velocity ($F\cos\theta$) contributes to the power being delivered to the object.
• If the force is in the same direction as the velocity ($\theta = 0^\circ$), $P = Fv\cos(0^\circ) = Fv$. The force is doing positive work and increasing the object's energy.
• If the force is opposite to the direction of velocity ($\theta = 180^\circ$), $P = Fv\cos(180^\circ) = -Fv$. The force is doing negative work and decreasing the object's energy (e.g., braking force, friction).
• If the force is perpendicular to the velocity ($\theta = 90^\circ$), $P = Fv\cos(90^\circ) = 0$. The force is doing no work and does not change the object's speed (e.g., centripetal force on an object moving in a circle at constant speed).

This formula is particularly useful when dealing with situations where force and velocity are known vectors, or when power output is related to the speed of a system (e.g., power of an engine needed to maintain a certain speed against resistive forces).

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