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Chapter 1: Problem 28

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

Expert verified
(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

01

(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.
02

(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.
03

(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.

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