δu_g = -w

Definition

The expression δu_g = -w defines the relationship between the change in gravitational potential energy (δu_g) of an object and the work done (w) on or by that object in a gravitational field. This equation highlights that the change in gravitational potential energy is equal to the negative of the work done by gravity, indicating that as an object gains potential energy while moving against gravity, work is done on it, and vice versa. This relationship emphasizes the conservation of energy principle within gravitational interactions.

5 Must Know Facts For Your Next Test

1. The negative sign in δu_g = -w indicates that when work is done against gravity, gravitational potential energy increases.
2. When an object falls freely under the influence of gravity, it loses potential energy equal to the work done by gravity.
3. This equation is fundamental in analyzing energy transformations in systems involving gravitational forces, such as roller coasters or pendulums.
4. If no external forces are acting on an object other than gravity, then the total mechanical energy (kinetic + potential) remains constant throughout its motion.
5. Understanding this relationship helps in solving problems related to height changes and work done in gravitational fields.

Review Questions

• How does δu_g = -w apply when lifting an object vertically against gravity?
  • When you lift an object vertically against gravity, you do work on it. According to δu_g = -w, the work you do is positive as you exert an upward force. This results in a gain of gravitational potential energy for the object. Therefore, as you increase its height, the change in gravitational potential energy is equal to the amount of work you performed against the gravitational force.
• Discuss how this equation illustrates the concept of conservation of energy within a gravitational system.
  • The equation δu_g = -w illustrates conservation of energy by showing that energy can be transformed from one form to another but remains constant within a closed system. When an object moves in a gravitational field, any increase in gravitational potential energy (as seen in lifting) must correspond to an equal amount of work done. Conversely, when falling, potential energy is converted into kinetic energy without any loss in total mechanical energy, demonstrating that energy is conserved.
• Evaluate a scenario where an object is thrown upwards and explain how δu_g = -w helps analyze its motion.
  • When an object is thrown upwards, it momentarily stops at its highest point before falling back down. During its ascent, work is done against gravity, resulting in an increase in gravitational potential energy as per δu_g = -w. The initial kinetic energy provided to throw the object converts into potential energy until it reaches maximum height. As it falls back down, this potential energy decreases while kinetic energy increases. This cyclical process exemplifies how this equation assists in analyzing motion and energy transformation throughout its trajectory.

Related terms

Gravitational Potential Energy: The energy stored in an object due to its position in a gravitational field, calculated as u_g = mgh, where m is mass, g is acceleration due to gravity, and h is height.

Work-Energy Theorem: A principle stating that the work done on an object is equal to the change in its kinetic energy, linking force, displacement, and energy transformations.

Conservative Forces: Forces that do not dissipate mechanical energy, meaning that the work done by or against these forces depends only on the initial and final positions, not on the path taken.