Question

For each of the following processes, does the potential energy of the object(s) increase or decrease? (a) The charge of two oppositely charged particles is increased. (b) \(\mathrm{H}_{2} \mathrm{O}\) molecule is split into two oppositely charged ions, \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-} .\) (c) A person skydives from a height of 600 meters.

Short answer

(a) The potential energy of the two oppositely charged particles increases. (b) The potential energy of the system when a water molecule is split increases. (c) The potential energy of a person skydiving decreases.

Step-by-step solution

(a) Determine the change in potential energy of two oppositely charged particles

In this case, the potential energy we're dealing with is electric potential energy. The formula for electric potential energy between two oppositely charged particles is given by: \[ U = \dfrac{kq_1q_2}{r} \] where \(U\) is the electric potential energy, \(k\) is Coulomb's constant, \(q_1\) and \(q_2\) are the magnitudes of the charges, and \(r\) is the distance between them. If the charge of the oppositely charged particles is increased while keeping the distance the same, then \(q_1\) and \(q_2\) in the formula would be larger. This means that \(U\) will also increase. Therefore, the potential energy of the two oppositely charged particles increases.

(b) Determine the change in potential energy when a water molecule is split

In this case, the potential energy involved is chemical potential energy. When a water molecule (\(\mathrm{H}_{2}\mathrm{O}\)) is split into two oppositely charged ions, \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\), energy is required to break the chemical bonds between the atoms. This energy is spent to overcome the attractive forces between the atoms. As a result, the potential energy of the system increases.

(c) Determine the change in potential energy of a person skydiving

In this case, we are dealing with gravitational potential energy. Gravitational potential energy can be defined as: \[ U_g = mgh \] where \(U_g\) is the gravitational potential energy, \(m\) is the mass of the person, \(g\) is the acceleration due to gravity, and \(h\) is the height above the ground. As the person skydives from a height of 600 meters, the height \(h\) decreases. Since the mass of the person \(m\) and acceleration due to gravity \(g\) remain constant, this means the gravitational potential energy \(U_g\) is decreasing as well. Therefore, the potential energy of a person skydiving decreases.

Key concepts

Electric Potential Energy

Electric potential energy is a form of potential energy that arises within electrical fields due to the interaction between electrically charged particles. When dealing with electric potential energy, one must consider Coulomb's Law, which describes the force between two charges. This force can be attractive or repulsive, depending on whether the charges are opposite or like respectively. In the scenario where we have two oppositely charged particles, electric potential energy is determined by the formula: \[ U = \dfrac{kq_1q_2}{r} \]Here, \( U \) is the electric potential energy, \( k \) is Coulomb’s constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) represents the distance between them.

• If the magnitude of either or both charges \(q_1\) and \(q_2\) increases, the electric potential energy \(U\) increases as well.
• It’s essentially a measure of the work that would be done to bring the charges from infinity to a distance \(r\) apart.

Remember that positive and negative charges attract, leading to changes in potential energy dependent on their magnitudes and distance.

Chemical Potential Energy

Chemical potential energy is the energy stored within the chemical bonds of a substance. This form of energy is crucial for driving chemical reactions, particularly the making and breaking of bonds between atoms. In a chemical reaction, changes in chemical potential energy occur when bonds are formed or broken.For example, when a water molecule (\(\mathrm{H}_{2} \mathrm{O}\)) is split into hydrogen ions (\(\mathrm{H}^{+}\)) and hydroxide ions (\(\mathrm{OH}^{-}\)), chemical potential energy is involved. This process requires energy to break the bonds within the water molecule:

• Breaking chemical bonds consumes energy, causing an increase in the system’s potential energy.
• This energy is required to overcome the attractive forces between the hydrogen and oxygen atoms.

Chemical reactions, such as splitting water into its ions, highlight the importance of chemical potential energy as it pertains to changes in a substance's internal energy.

Gravitational Potential Energy

Gravitational potential energy refers to the energy an object possesses due to its position in a gravitational field, primarily dependent on the object’s height above a reference point, usually Earth’s surface. The formula for this type of potential energy is given by:\[ U_g = mgh \]In this equation, \(U_g\) is the gravitational potential energy, \(m\) is the mass of the object, \(g\) is the acceleration due to gravity (approximated as \(9.8 \, m/s^2\) on Earth), and \(h\) is the height above a reference level.

• As the height \(h\) increases, so does the gravitational potential energy \(U_g\).
• When an object, like a skydiver, descends, its height decrease results in a decrease in gravitational potential energy.

Thus, as a skydiver jumps from a height of 600 meters, the reduction in height translates to a reduction in gravitational potential energy until they reach the ground.