91Ó°ÊÓ

Gravitational potential energy is an important concept in physics and defines the energy an object possesses due to its position within a gravitational field. In simpler terms, it is the energy stored in an object as a result of its height above the ground. A higher object has more gravitational potential energy.

The key formula to calculate gravitational potential energy ( U ) is:

• \( \Delta U = mgh \)

where:

• \( m \) is the mass of the object,
• \( g \) represents the acceleration due to gravity (typically \(9.8 \text{ m/s}^2\) on Earth),
• \( h \) is the height from which an object is raised.

In the exercise, the elevator with a mass of \(1200 \text{ kg}\) moves upward a distance of \(54 \text{ m}\). By applying the formula, we find the change in gravitational potential energy to be \(635040 \text{ J}\). Understanding these basics allows physicists to determine how much energy is required to change the state of motion of objects.

Work done in physics essentially tells us how much energy is transferred when a force acts over a distance. It is integral in understanding how energy is being utilized or absorbed by systems. To calculate work done ( W ) in our scenario, we consider not only lifting the elevator but counteracting the counterweight's effect.

The general formula for calculating work is:

• \( W = F \times d \)

For the elevator, an interesting twist is added: the effective force due to the counterweight. The effective mass to be lifted is the elevator mass minus the counterweight mass, leading to:

• \( m_{\text{eff}} = m_{\text{elevator}} - m_{\text{counterweight}} \)
• \( m_{\text{eff}} = 250 \text{ kg} \)

This reduces the effective work needed by the motor:

• \( W = m_{\text{eff}} \times g \times h = 132300 \text{ J} \)

Knowing the work done helps us later when computing power, offering insight into the mechanical energy required by the system.

Power is the rate at which work is done over time, specifically how quickly energy is transformed or transferred in a system. Average power provides a good overview of how effective the energy conversion is over a specific period.

To calculate power ( P ) in this context, apply the formula:

• \( P = \frac{W}{t} \)

where:

• \( W \) is the work done,
• \( t \) is the time over which the work is done.

For the elevator, with work done \( W = 132300 \text{ J} \) and time taken \( t = 180 \text{ s} \), the average power is computed as:

• \( P = \frac{132300 \text{ J}}{180 \text{ s}} \approx 735 \text{ W} \)

The concept of power can be related to everyday scenarios, like how a car engine's power tells us how much energy can be provided to get the car moving. Here it shows how the motor efficiently lifts the elevator with a demand for energy conversion over time.