A 200 g air-track glider is attached to a spring. The glider is pushed in 10 cm and released. A student with a stopwatch finds that 10 oscillations take 12.0 s. What is the spring constant?

Verified step by step guidance

1. Identify the mass of the glider (m) and convert it to kilograms if necessary. In this case, the mass is 200 g, which is 0.2 kg.
2. Determine the displacement from equilibrium (x), which is given as 10 cm. Convert this to meters by dividing by 100, so x = 0.1 m.
3. Calculate the period of one oscillation (T) by dividing the total time for 10 oscillations by 10. Given that 10 oscillations take 12.0 s, the period T for one oscillation is 12.0 s / 10 = 1.2 s.
4. Use the formula for the period of a mass-spring system, T = 2\pi \sqrt{\frac{m}{k}}, where k is the spring constant. Rearrange this formula to solve for k: k = \frac{4\pi^2 m}{T^2}.
5. Substitute the values of m and T into the rearranged formula to find the spring constant k.

Key Concepts

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 as F = -kx, where F is the force, k is the spring constant, and x is the displacement. This principle is fundamental in understanding how springs behave when compressed or stretched.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In the case of a spring-mass system, the motion is characterized by a restoring force proportional to the displacement, leading to a sinusoidal motion. The period of oscillation depends on the mass and the spring constant.

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 and k is the spring constant. This relationship is crucial for determining the spring constant when the period is known.