Final answer:
The force constant of the spring is primarily responsible for the vertical oscillations of the clown in a jack-in-the-box toy, as it determines the stiffness and restoring force of the spring.
Step-by-step explanation:
The physical property primarily responsible for the vertical oscillations of the clown in a jack-in-the-box toy is the force constant of the spring. The force constant (also known as the spring constant) is a measure of the stiffness of a spring. According to Hooke's law, the force exerted by a spring is proportional to its displacement, which is why the force constant of the spring determines how the toy clown oscillates: the greater the force constant, the more force it can exert for a given displacement, leading to the characteristic oscillation when the spring is released.