Electrical Energy and Gravitational Potential Energy Calculations

A detailed exploration of energy transfer in a motor-driven mass lifting task

Highlights

Electrical Energy Calculation: Determined by multiplying voltage, current, and time.
Gravitational Potential Energy (GPE): Given by the formula GPE = m × g × h.
Energy Losses: Discrepancy between electrical energy transferred and GPE gained is due to losses like friction and heat.

Detailed Analysis and Calculations

(i) Electrical Energy Transferred to the Motor

Calculation Process

The electrical energy provided to a motor is determined by the power delivered to it, multiplied by the time for which the power is supplied. This can be expressed mathematically as:

\( E = V \times I \times t \)

Where:

\( V \) is the voltage (in volts, V),
\( I \) is the current (in amperes, A),
\( t \) is the time (in seconds, s).

Application of the Formula

Given the problem:

Voltage, \( V = 12 \, \text{V} \)
Current, \( I = 2.1 \, \text{A} \)
Time, \( t = 1.5 \, \text{s} \)

Substitute these values into the formula:

\( E = 12 \times 2.1 \times 1.5 \)

This gives:

\( E = 37.8 \, \text{J} \)

When rounded to two significant figures, the electrical energy transferred is approximately 38 J.

(ii) Equation Linking Gravitational Potential Energy

Fundamental Relationship

The gravitational potential energy (GPE) of an object at a certain height is given by the equation:

\( \text{GPE} = m \times g \times h \)

Where:

\( m \) is the mass (in kilograms, kg),
\( g \) is the acceleration due to gravity (in meters per second squared, m/s²),
\( h \) is the height (in meters, m).

(iii) Gravitational Potential Energy Gained by the 130g Mass

Step-by-Step Calculation

To find the gravitational potential energy gained by the mass, follow these steps:

Convert the mass: The given mass is 130 g. Converting this into kilograms:
\( 130 \, \text{g} = 0.130 \, \text{kg} \)
Convert the height: The height provided is 63 cm. In meters, this is:
\( 63 \, \text{cm} = 0.63 \, \text{m} \)
Use standard value of gravity: For calculations on Earth, \( g \) is usually taken as \( 9.81 \, \text{m/s}^2 \).
Substitute into the GPE formula:
\( \text{GPE} = 0.130 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 0.63 \, \text{m} \)

Multiplying these values:

\( \text{GPE} \approx 0.130 \times 9.81 \times 0.63 \approx 0.803 \, \text{J} \)

Rounded to two significant figures, the gravitational potential energy is approximately 0.80 J.

(iv) Discrepancy Between Electrical Energy and GPE

Understanding the Energy Losses

When analyzing the process of lifting a mass using a motor, it is essential to realize that the electrical energy provided to the system is not entirely converted into gravitational potential energy. Here are the key factors responsible for the discrepancy:

Motor Inefficiencies: Not all of the electrical energy is converted into mechanical work. Some energy is dissipated due to internal inefficiencies of the motor's design and operation.
Frictional Losses: Friction occurs within the moving parts of the motor and the lifting mechanism. This friction converts part of the energy into heat, reducing the effective energy available for performing work.
Heat Generation: Electrical resistance both in the motor windings and associated circuitry leads to heat production, which is an energy loss mechanism.
Sound Energy: Some of the input energy is radiated away as sound when the motor operates, although this is typically a minor contributor compared to other losses.
Other System Losses: Additional energy losses can occur in the form of vibrations, air resistance, and inefficiencies in the gear or pulley systems used to lift the mass.

All these losses result in the gravitational potential energy (0.80 J) being less than the total electrical energy (38 J) delivered to the motor.

Supplementary Data and Comparison Table

Summary of Calculations

Parameter	Value	Unit	Calculation
Voltage (\(V\))	12	V	Given
Current (\(I\))	2.1	A	Given
Time (\(t\))	1.5	s	Given
Electrical Energy (\(E\))	\(12 \times 2.1 \times 1.5 = 37.8\) → 38	J	\( E = V \times I \times t \)
Mass (\(m\))	0.130	kg	\(130 \, \text{g} = 0.130 \, \text{kg}\)
Height (\(h\))	0.63	m	\(63 \, \text{cm} = 0.63 \, \text{m}\)
Acceleration (\(g\))	9.81	\(m/s^2\)	Standard value
Gravitational Potential Energy (GPE)	\(0.130 \times 9.81 \times 0.63 \approx 0.80\)	J	\( \text{GPE} = m \times g \times h \)

In-Depth Discussion

Relevance of the Calculations

The problem presented serves as an example of how energy conversion is far from ideal in practical applications. While the electrical energy transferred to the motor is a straightforward product of voltage, current, and time, the work performed by the motor in lifting a mass illustrates a more complex interaction where only a fraction of the energy is utilized for the intended mechanical task.

This difference is a direct demonstration of the law of conservation of energy, where the energy that is not converted into useful mechanical energy is transformed into forms such as heat or acoustic energy. In engineering systems and physics experiments, understanding and quantifying these losses is critical in designing more efficient machines and in improving the theoretical models that predict system behavior.

Examining the Role of Inefficiencies in Real-World Systems

Motor Efficiency

Motors, whether they are used in industrial applications or simple educational experiments, seldom operate at 100% efficiency. Several factors contribute to the losses observed:

Electrical Resistance: The wires and components in the motor have inherent resistance, which reduces the effective power output as some energy is lost as heat.
Mechanical Friction: Bearings, brushes in commutator motors, and the contact surfaces between moving parts result in frictional losses. Advances in lubricants and engineering design can reduce these losses, but they cannot be entirely eliminated.
Heat Dissipation: Inefficient energy conversion in the motor's coils and other components causes temperature rises, leading to thermal energy being radiated away.
Energy Conversion Rates: The conversion process of electrical energy into mechanical energy is not instantaneous. There might be periods during transient operation where less than the total power is converted per unit time.

Impact on System Design

Engineers and designers must account for these losses when developing systems. By improving insulation, using higher quality materials, and designing better moving components, improvements in efficiency can be made. However, even in systems where efficiency is maximized, factors like friction and heat losses still ensure the output energy (in this case, the gravitational potential energy) remains lower than the input electrical energy.

Practical Implications

Understanding these discrepancies is not only valuable from a theoretical standpoint but has practical implications as well. In applications ranging from industrial lift systems to electric vehicles, energy management and efficiency are critical design criteria. Improving these factors can lead to significant energy savings, reduced wear and tear on mechanical components, and extended system lifespans.

An example of practical importance can be seen in energy audits for machinery, where engineers measure both the energy input and the effective work output. Such analyses help in identifying areas where inefficiencies are most pronounced, guiding targeted improvements in equipment design and maintenance schedules.

Conclusion and Final Thoughts

This examination offers a comprehensive analysis of the motor-driven mass-lifting scenario. The electrical energy transferred to the motor is calculated using the formula \( E = V \times I \times t \) and is approximately 38 J when using the provided parameters. In parallel, the gravitational potential energy gained by a 130 g mass lifted by 63 cm is determined using \( \text{GPE} = m \times g \times h \), which comes out to nearly 0.80 J.

The substantial difference between these two energy values is attributed to inherent system inefficiencies. Losses due to friction, internal resistance, heat generation, and other processes ensure that only a fraction of the electrical energy contributes to increasing the gravitational potential energy of the mass. This scenario highlights fundamental principles in energy conservation and efficiency that are critical in both educational contexts and real-world engineering applications.

By analyzing and quantifying these factors, one gains essential insights into why practical systems stray from idealized models and what strategies can be adopted to enhance performance. It also underscores the importance of careful energy management in designing devices, as even small inefficiencies can have significant cumulative effects over time.