Power (HSC SSCE Physics): Revision Notes

Power

Introduction to power

Imagine two cranes that can both lift a $10$ tonne block of concrete to the top of a building. The first crane completes this task in $1$ minute, while the second crane takes $2$ minutes. Although both cranes do the same amount of work on the load, they take different amounts of time. The crane that completes the task faster is more powerful.

What is power?

Power is the rate at which work is done. Since work represents the energy transferred by a force, we can also define power as the rate at which energy is transferred or transformed from one form to another.

The mathematical definition of power is:

$P = \frac{\Delta W}{\Delta t} = \frac{\Delta E}{\Delta t}$

Where:

• $P$ is power
• $\Delta W$ is the work done
• $\Delta E$ is the change in energy
• $\Delta t$ is the time interval

Units of power

The SI unit of power is the watt (W), named after James Watt. One watt is equivalent to:

$1\text{ W} = 1\text{ J s}^{-1} = 1\text{ N m s}^{-1}$

This means that one watt represents the transfer of one joule of energy per second.

Energy and power relationships

We can rearrange the power equation to find the energy transferred or transformed in a process:

$\Delta E = P\Delta t$

This equation tells us that the total energy change equals the power multiplied by the time interval over which the power is applied.

Graphical representations

The relationship between power, energy, and time can be visualized using graphs.

Relating power to force and velocity

We can derive another useful expression for power by connecting it to force and velocity. Starting with the work equation:

$W = F_{\parallel}s = Fs\cos\theta$

Substituting this into the power equation:

$P = \frac{\Delta E}{\Delta t} = \frac{W}{\Delta t} = \frac{Fs\cos\theta}{\Delta t} = F\cos\theta \frac{\Delta s}{\Delta t} = Fv\cos\theta$

Therefore:

$P = Fv\cos\theta$

Worked example: crane lifting a concrete block

Factors affecting power in vehicles

The top speed of a car is determined by several factors, with engine power being a key consideration. Understanding the forces that oppose motion helps explain why more power is needed to maintain higher speeds.

Air resistance

As a car's speed increases, air resistance increases significantly. This force does negative work on the car, removing energy from the system. To maintain a constant speed, the engine must supply enough power to replace this lost energy by converting fuel energy. This is why fuel consumption increases dramatically at higher speeds.

Rolling friction

Rolling friction acts on the tyres as they move along the road surface. In an ideal scenario, a wheel would roll without any slipping against the ground. However, in reality:

• There may be slight slipping between the tyre and road surface
• Both the tyre and road surface undergo deformation (change shape temporarily)

These factors combine to create rolling friction, which must be overcome by the engine to keep the car moving at constant speed.

Internal friction

Friction occurs at all points along the drive train of a car (the system that transmits power from the engine to the wheels). This internal friction reduces the efficiency of the vehicle by decreasing the force that the tyres can apply to the ground. Less of the engine's power actually reaches the wheels to propel the car forward.

Investigation: power transfer

This practical investigation explores how energy transforms from gravitational potential energy to kinetic energy, and how to measure the rate of this energy transfer (power).

Aim

To investigate how power is transferred from falling weights to a toy car or block, and how friction affects this energy transfer.

Materials

• Pulley attached to table edge
• String
• Masses and mass holder
• Tape measure
• Tape
• Toy cars
• Blocks with different surfaces (smooth and rough)
• Stopwatch or motion sensor with data logger

Risk assessment

Method

Setup:

1. Measure and record the masses of the toy car, blocks, and falling weights
2. Set up your equipment as shown in the diagram, with the toy car lined up directly toward the pulley
3. Mark start and finish lines on the table using tape
4. Set up data-logging equipment to record the car's motion, or prepare a stopwatch
5. Place the car at the start line and allow the weights to fall freely through a measured distance
6. Record the motion or time how long it takes to reach the finish line

Variations to test:

1. Lock the wheels so they cannot rotate (using tape) and repeat
2. Repeat with blocks having various surface textures

Results

Record your measurements in a table:

Object being pulled	Time for fall (s)	Final speed (m s⁻¹)	Change in $E_k$ (J)	Mass of falling weights (kg)	Distance fallen (m)	Change in $U_g$ (J)	Change in $E_k$ (J)
Car – rolling
Car – sliding
Rough block – sliding
Smooth block – sliding

Analysis

Perform these calculations for each measurement:

1. Calculate the final kinetic energy of the car/block: $E_k = \frac{1}{2}m_{\text{car}}v^2$
2. Calculate the power transferred to the car: $P_{\text{to car}} = \frac{\Delta E_k}{\Delta t}$
3. Calculate the change in gravitational potential energy of the falling weights: $\Delta U_g = m_{\text{weight}}g\Delta h$
4. Calculate the final kinetic energy of the weights: $E_k = \frac{1}{2}m_{\text{weights}}v^2$(Note: The weights and car move at the same speed because they're connected by the string)
5. Calculate the power transformed into kinetic energy of the weights: $P_{\text{to weights}} = \frac{\Delta E_k}{\Delta t}$
6. Calculate the power transferred from the weights: $P_{\text{from weights}} = \frac{\Delta U_g}{\Delta t}$
7. Calculate the difference between the power from potential energy and the combined kinetic energy. This difference represents the rate at which energy is lost as thermal energy due to friction.

Discussion

Consider these questions:

1. For each rolling and sliding situation, how much of the power from the falling weights increased the kinetic energy of the car/block?
2. Was there a difference between sliding and rolling? What evidence is there for rolling resistance? How did kinetic friction affect sliding objects?
3. Was mechanical energy conserved in any of these situations? Explain your reasoning.

Conclusion

Based on your data and analysis, write a conclusion that answers your inquiry question about power transfer and the effects of friction on energy transformation.