Question

A person gets into an elevator at the lobby level of a hotel together with his 30 -kg suitcase, and gets out at the 10th floor \(35 \mathrm{m}\) above. Determine the amount of energy consumed by the motor of the elevator that is now stored in the suitcase.

Short answer

Answer: 10251 J (Joules)

Step-by-step solution

Identify the given variables

We are given the following information: -Mass of the suitcase (m): 30 kg -Vertical distance covered (h): 35 m We also know the value of acceleration due to gravity (g): 9.81 m/s².

Calculate the gravitational potential energy gained

The potential energy gained by the suitcase can be found using the formula: PE = m * g * h. Substitute the given values: PE = (30 kg) * (9.81 m/s²) * (35 m).

Evaluate the expression

Multiply the values to get the potential energy gained: PE = (30 kg) * (9.81 m/s²) * (35 m) = 10251 J(Joules).

State the final answer

The amount of energy consumed by the elevator motor and now stored in the suitcase as gravitational potential energy is 10251 J (Joules).

Key concepts

Work and Energy in Physics

Understanding 'work' and 'energy' is fundamental in physics, especially when dealing with the conservation of energy in mechanical systems.

Work in physics is defined as the product of force and displacement in the direction of the force. Mathematically, it's expressed as \( W = F \times d \), where \(W\) is work, \(F\) is force, and \(d\) is displacement. The unit of work is the Joule (J), which is also the unit of energy.

Energy is the capacity to do work. There are various forms of energy, including kinetic energy (energy of motion), potential energy (stored energy), and others like thermal and chemical energy. Gravitational potential energy, as in our elevator problem, is the energy an object possesses due to its position in a gravitational field. The formula for gravitational potential energy is \( PE = m \times g \times h \), where \(m\) is mass, \(g\) is the acceleration due to gravity, and \(h\) is the height above a reference point.

This formula shows us that the energy transferred to the suitcase is because of work done by the elevator against Earth's gravity. This energy is now stored as potential energy in the suitcase.

Elevator Problem Physics

Elevator problems are a classic example of how the principles of work and energy are applied in real-world scenarios. Analyzing an elevator's motion involves understanding the forces at play and the energy transformation that occurs.

An elevator's motor does work to lift objects against the force of gravity. When the motor raises a mass, such as a suitcase, it increases the mass's gravitational potential energy. The amount of work the motor has to do is equal to the increase in the potential energy of the mass. If the question involves how much energy the motor consumes, we must account for efficiencies or losses, but in simpler classroom problems, we often assume an ideal situation.

In the exercise provided, the solution calculates the straightforward increase in potential energy of the suitcase. However, in a more complex case, you might need to consider additional factors like the kinetic energy if the elevator is moving at different speeds or the energy lost to friction.

Mechanical Energy Conservation

The principle of mechanical energy conservation is a powerful tool in solving physics problems. It states that if no external work is done by non-conservative forces (like friction), the total mechanical energy of a system remains constant.

Mechanical energy is the sum of kinetic and potential energy within a system. When an object moves higher in a gravitational field, its potential energy increases while kinetic energy might decrease if it slows down, but the total energy remains the same.

Application in Elevator Systems

In a frictionless scenario, the work done by the elevator's motor would convert entirely into potential energy of the lifted object, exemplified by our suitcase.

Understanding that the total mechanical energy before and after the lift must be equal allows us to analyze and solve various problems. In real-life applications, factors such as air resistance and friction will cause some of the motor's work to be transformed into thermal energy, which should be accounted for in a thorough analysis of the system.