Performance Task

Newton’s law of universal gravitation is

where $G=6.67\times {10}^{-11}{\text{m}}^3\text{/kg}\cdot {\text{s}}^2$ . This describes the gravitational force between two point masses m₁ and m₂.

Coulomb’s law is

where $k=8.99\times {10}^9\text{N}\cdot {\text{m}}^{\text{2}}{\text{/C}}^{\text{2}}$ . This describes the electric force between two point charges q₁ and q₂.

(a) Describe how the force in each case depends on the distance r between the objects. How do the forces change if the distance is reduced by half? If the distance is doubled?

(b) Describe the similarities and differences between the two laws. Consider the signs of the quantities that create the interaction (i.e., mass and charge), the constants G and k, and their dependence on separation r.

(c) Given that the electric force is much stronger than the gravitational force, discuss why the law for gravitational force was discovered much earlier than the law for electric force.

(d) Consider a hydrogen atom, which is a single proton orbited by a single electron. The electric force holds the electron and proton together so that the hydrogen atom has a radius of about $0.5\times {10}^{\mathrm{-10}}\text{m}$ . Assuming the force between electron and proton does not change, what would be the approximate radius of the hydrogen atom if $k=6.67\times {10}^{\mathrm{-11}}\text{N}\cdot {\text{m}}^{\text{2}}{\text{/C}}^{\text{2}}$ ?