7 - Power and Energy Applications

1. Set Up the Energy Model

Energy methods begin by deciding what belongs to the system and comparing its initial and final states. The system could be a single object, an object together with Earth, or an object plus a machine and its surroundings.

A useful accounting equation is

Here, $K$ is kinetic energy, $U$ represents stored potential energy, $\Delta E_{\text{th}}$ is the increase in thermal energy, and $W_{\text{ext}}$ is work done by forces outside the system. For an isolated system, $W_{\text{ext}}=0$, so total energy is conserved.

For motion near Earth’s surface, gravitational potential energy is $U_g=mgy$. For a spring, elastic potential energy is $U_s=\frac{1}{2}kx^2$. The zero level for gravitational potential energy can be selected freely; only changes in that energy matter.

Takeaway: Define the system before choosing an equation, and represent friction as a transfer into thermal energy rather than as destroyed energy.

2. Relate Energy and Power

Power describes how quickly energy is transferred or converted, whereas energy describes the amount transferred. The average rate is

The SI unit is the watt:

Thus, a device rated at $500\ \text{W}$ transfers or converts energy at a rate of $500\ \text{J/s}$ under the stated conditions. For constant power, the transferred energy is

When power changes with time, total energy is the area under the power-versus-time graph:

Takeaway: Keep energy and power conceptually separate: joules measure an amount, while watts measure an amount per unit time.

3. Calculate Power from Force and Motion

The Instantaneous power delivered by a force depends on the component of the force along the velocity:

Equivalently, if $F_{\parallel}$ is the force component parallel to the velocity, then

Important cases are:

• If the force is parallel to the velocity, $P=Fv$.

• If the force is opposite the velocity, $P=-Fv$, so the force removes mechanical energy.

• If the force is perpendicular to the velocity, $P=0$, so it does no instantaneous work.

For an object lifted upward at constant speed, the lifting force approximately balances the weight. The mechanical power delivered to the object is therefore

This is the useful mechanical power. A motor with less than perfect efficiency requires greater input power.

Takeaway: Use $P=Fv$ only when the force is parallel to the velocity; otherwise include the angle or use the parallel component.

4. Evaluate Efficiency

Efficiency compares the desired output with the total input:

or, during steady operation,

As a percentage,

For an ordinary machine, $0<\eta<1$. The input power can be found from

and useful output energy can be found from

The part of the input that is not useful output may become thermal energy, sound, vibration, or another unintended form. It has not disappeared.

Example: If a motor provides $3.68\ \text{kW}$ of useful power while receiving $5.0\ \text{kW}$, then

Takeaway: An inefficient machine must receive more input power than the useful power it delivers.

5. Apply Power to Lifting and Friction

For a lifting problem, the useful output is often the increase in gravitational potential energy. A $250\ \text{kg}$ load lifted through $12\ \text{m}$ in $8.0\ \text{s}$ has useful output power

Therefore, $P_{\text{useful}}\approx3.7\ \text{kW}$. The average upward speed is $v=h/\Delta t=1.5\ \text{m/s}$, which gives the same result through $P=mgv$.

For a car traveling at constant speed, the engine can still transfer energy even though the net force is zero. If a car moves at $20\ \text{m/s}$ against a resistive force of $600\ \text{N}$, the engine supplies

The supplied energy is transferred primarily into thermal energy in the tires, road, air, and engine components.

Takeaway: Constant speed means zero net force and zero net work on the object, not zero power from the engine.

6. Use Energy Accounting with Friction

An energy balance is especially useful when friction is present. Consider a $2.0\ \text{kg}$ block that starts from rest at a height of $3.0\ \text{m}$, with friction transferring $12\ \text{J}$ into thermal energy. Choose the block–Earth system and take the bottom as $y=0$. Since there is no external work,

The initial kinetic energy is zero, so

Thus,

Substituting the values gives

so

Without friction, more gravitational potential energy would become kinetic energy and the final speed would be higher.

Takeaway: Include frictional energy explicitly as an increase in thermal energy; do not simply remove energy without representing where it goes.

7. Read Power–Time Graphs

A power-versus-time graph represents a rate, so its area represents transferred energy. For a machine operating at $800\ \text{W}$ for $5.0\ \text{s}$ and then at $300\ \text{W}$ for $10\ \text{s}$, the total input energy is

The total time is $15\ \text{s}$, so the average power is

This average is not the same as the power during either individual interval. For a graph with several segments, calculate the area of each segment and add the results before dividing by the full time interval.

Takeaway: Find total energy from graph area, then find average power by dividing that total energy by total time.

8. Solve and Check Applications

Use the following sequence for unfamiliar problems:

1. Define the system. Decide which objects and energy stores are included.

2. Identify the states. Record initial and final speeds, heights, spring compression, and time intervals.

3. List transfers. Include external work, kinetic energy, potential energy, and thermal energy.

4. Select the equation. Use $P=\Delta E/\Delta t$, $P=Fv\cos\theta$, an efficiency relation, or an energy balance.

5. Convert units. Use seconds for time, meters for distance, watts for power, and consistent energy units.

6. Check the result. Power should have units of watts, efficiency should be dimensionless or a percentage, and an inefficient machine should require more input power than useful output power.

Common checks:

• Do not confuse power with energy.

• Do not use $P=Fv$ when the force is not parallel to the velocity.

• Do not infer that constant speed means no energy transfer.

• Do not accept an efficiency greater than $100\%$ without finding a system, input, or unit error.

• Do not omit the thermal-energy increase caused by friction.

Final takeaway: A clear system boundary, a complete energy account, and careful attention to units connect power, energy transfer, friction, and efficiency into one consistent method.

• Power–time graph