In a physics lab, you attach a 0.200-kg air-track glider to the end of an ideal spring of negligible mass and start it oscillating. The elapsed time from when the glider first moves through the equilibrium point to the second time it moves through that point is 2.60 s. Find the spring's force constant.

Verified step by step guidance

1. Identify the given values: mass of the glider (m) = 0.200 kg, and the time for one complete oscillation (T) = 2.60 s.
2. Recognize that the time for one complete oscillation (T) is the period of the spring-mass system. Since the problem states the time from the first to the second pass through the equilibrium point, this time represents half of a complete oscillation. Therefore, multiply the given time by 2 to find the period of the oscillation.
3. Use the formula for the period of a mass-spring system, which is given by T = 2\pi \sqrt{\frac{m}{k}}, where T is the period, m is the mass, and k is the spring constant.
4. Rearrange the formula to solve for the spring constant k: k = \frac{4\pi^2 m}{T^2}.
5. Substitute the values of m and T into the rearranged formula to calculate the spring constant k.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In this case, the air-track glider attached to the spring exhibits SHM, characterized by a restoring force proportional to the displacement from equilibrium. The time taken for one complete cycle is known as the period, which is crucial for determining the spring's properties.

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, expressed mathematically as F = -kx, where F is the force, k is the spring constant, and x is the displacement. This law is fundamental in understanding how springs behave and is essential for calculating the spring's force constant in the given problem.

Period of Oscillation

The period of oscillation is the time taken for one complete cycle of motion in a harmonic oscillator. For a mass-spring system, the period (T) can be calculated using the formula T = 2π√(m/k), where m is the mass attached to the spring and k is the spring constant. In this scenario, knowing the elapsed time for the glider to pass through the equilibrium point helps in determining the spring's force constant.