Question

A \(10 \mathrm{~kg}\) box is lifted \(2 \mathrm{~m}\) off the floor and placed on a frictionless horizontal conveyor to take it \(30 \mathrm{~m}\) across a warehouse. At the end of the conveyor, it is lowered \(1 \mathrm{~m}\) where it ends up on a shelf. How much net gravitational potential energy was given to the box from the start to the end of the process?

Short answer

Answer: The net gravitational potential energy given to the box throughout the process is -98.1 J.

Step-by-step solution

Determine the initial and final heights

Initially, the box is lifted 2 meters off the floor, so the initial height (h1) is 2 meters. After moving horizontally and lowered 1 meter, the final height (h2) of the box would be 1 meter (since it was lowered from the 2 meters initial height).

Calculate the change in gravitational potential energy

The change in gravitational potential energy (ΔPE) can be calculated using the formula: ΔPE = m * g * (h2 - h1) where m is the mass of the object (10 kg), g is the acceleration due to gravity (9.81 m/s²), h1 is the initial height (2 m), and h2 is the final height (1 m).

Solve for the net gravitational potential energy

Plugging the values into the formula, we get: ΔPE = 10 kg * 9.81 m/s² * (1 m - 2 m) ΔPE = 10 kg * 9.81 m/s² * (-1 m) ΔPE = -98.1 J The net gravitational potential energy given to the box is -98.1 J, which indicates a loss of potential energy as it is lowered to a final height of 1 meter.

Key concepts

Physics Education

Physics education is fundamental to understanding the way our world operates. It covers a wide range of topics, from the tiniest particles to the largest celestial bodies. A crucial part of physics education involves teaching about energy, including gravitational potential energy, as demonstrated in the exercise above.

Gravitational potential energy is the energy an object possesses because of its position in a gravitational field. The formula to calculate this energy is given by the expression:\[ PE = m \cdot g \cdot h \]where:

• PE is the gravitational potential energy
• m is the mass of the object
• g is the gravitational acceleration, approximately 9.81 m/s² on Earth
• h is the height of the object above the reference point

This exercise helps highlight how these principles are applied practically. It offers students an opportunity to use mathematical calculations to understand energy changes based on varying physical positions.

By methodically calculating the change in gravitational potential energy, students can better appreciate how energy is conserved and transformed in different scenarios.

Energy Conservation

Energy conservation is a fundamental principle in physics. It states that energy cannot be created or destroyed; it can only be changed from one form to another. When considering gravitational potential energy, this means analyzing how energy is stored and transferred within a system.

In the exercise example, the box gains gravitational potential energy when lifted to a height of 2 meters. This energy is stored as its height increases. As it moves horizontally, the gravitational potential energy remains unchanged since the height does not vary along this path. Finally, as the box is lowered onto the shelf, some of this stored energy is released.

The calculation shows a decrease in gravitational potential energy of \(-98.1\) Joules, signifying that energy has been transferred elsewhere in the system. This change in energy aligns with the concept of energy conservation, where the energy initially given to the box is effectively transformed as it is lowered to a new position. Ensuring that students grasp this concept is vital for understanding broader physics topics, including mechanical systems and energy efficiency.

Kinematics

Kinematics, another core area of physics, deals with the motion of objects without considering the forces that cause such motion. It involves understanding and analyzing various aspects like velocity, acceleration, displacement, and how these quantities interact with time.

In the given exercise, the box's movement involves vertical and horizontal displacements, both of which are integral parts of kinematics. The vertical displacement when lifted ( from the floor to 2 meters) and lowered ( from 2 meters down to 1 meter) affects the calculation of gravitational potential energy.

While the focus is on energy, understanding the box's motion helps clarify how different types of movement correspond to changes in potential energy. Movement in kinematics can occur without changes in energy as seen when the box travels horizontally.

Students can explore how these aspects are simplified in frictionless scenarios, making the exercises manageable while illustrating foundational physics principles. By connecting kinematics to potential energy calculations, students are encouraged to see the interconnections between different physics topics.