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?

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Convert the mass of the glider from grams to kilograms. Since 1 g = 0.001 kg, the mass \( m \) is \( 200 \times 0.001 = 0.2 \, \text{kg} \).

Determine the period of one oscillation. The period \( T \) is the time for one complete oscillation, which can be calculated by dividing the total time for 10 oscillations by 10: \( T = \frac{12.0}{10} \, \text{s} \).

Use the formula for the period of a mass-spring system: \( T = 2\pi \sqrt{\frac{m}{k}} \), where \( T \) is the period, \( m \) is the mass, and \( k \) is the spring constant. Rearrange this formula to solve for \( k \): \( k = \frac{4\pi^2 m}{T^2} \).

Substitute the known values into the formula: \( m = 0.2 \, \text{kg} \) and \( T \) (calculated in step 2). Ensure that \( T \) is squared in the denominator.

Simplify the expression to find the spring constant \( k \). The units of \( k \) will be in \( \text{N/m} \), as it represents the stiffness of the spring.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 mass-spring 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 system. 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. This relationship is crucial for determining the spring constant from the observed oscillation time.